Value of liquidity in financial markets

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Value of liquidity in financial markets
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Liquidity (Economics)   ( lcsh )
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Finance, Insurance and Real Estate thesis Ph.D
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Thesis (Ph. D.)--University of Florida, 1994.
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Includes bibliographical references (leaves 88-92).
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by Vinay Datar.
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Typescript.
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Vita.

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VALUE OF LIQUIDITY IN FINANCIAL MARKETS







By

VINAY DATAR


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1994













ACKNOWLEDGEMENTS

My sincere gratitude for the invaluable guidance, encouragement and


support of my dissertation committee cannot be sufficiently articulated.


This


intense effort would not have been possible without the understanding and

support from my family and I shall, forever, remain grateful.


















TABLE OF CONTENTS


ACKNOWLEDGMENTS


. .* .* .* .* .* .* 9 9 9 9 9 9 9 9 3 ii


ABSTRACT


CHAPTERS


A TUTORIAL ON LIQUIDITY


Introduction


* 3 1 3 9 9 9 9 9 9 9 9 9 9 9 9 1


. P 3 9 9 9 9 9 9 9 9 1


What


s Liquidity? Overview of Literature
Why do people demand liquidity? .
Why is it costly to supply liquidity?


* 9 9 9 9 3
. . 3


9 9 9 9 9 9 9 9 9 5


Incomplete markets .
Imperfect markets .
Market micro-structure
Different measures of liquidity
Value of Liquidity: A Basic Model


The economy


* 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 3 9 5/
* 9 9 9 S 9 9 9 9 9 9 9 3 9 9 9 9 #7
S 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8
S 9 9
* 9 9 a a 3 9 9 9 3 3 9 9 9 9 12


9 9 9 9 9 9. 9 3 9 9 9 9 9 9 9 9 9 9 9 9 9 9 1.


Price of the
Price of the
Discussion


'liquid'
'illiquid


asset


asset


a a 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 14.
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 91 &14


. 1


Summary


CROSS-SECTION OF STOCK RETURNS REVISITED


LIQUIDITY PREMIA AND ROLE OF SIZE


Introduction


. .. .. 23


S9 9 9 2


Data and Methodology


9 9 S 3 9 9 9 9 9 9 9 9 9 9 9 9 9 9 3 9 4 9 9 .2


Description of data
Estimation of beta
Proxies for liquidity


* 9 9 9 9 9
* 9 9 9 9 *


* 9 9 0 3
* 9 9 9 .9


* 9 9 2
* 9 9 2
* 9 9 2


1


_ _I 1 I









Summary


. S 40


IMPACT OF LIQUIDITY ON PREMIA/DISCOUNTS IN


CLOSED-END FUNDS


Introduction


a a a a a 50


a a a 50


Liquidity and Premia/Discounts . .
How is liquidity related to premia/discounts


Testable hypotheses
Proxies for liquidity


Data and Methodology
Data . .


* C C C U C 53
* C S 53


S S S S U C b C C S C S C S C S S C C C C 5 4I


. . 56


* C S . a a C S a a S S C C C C C C C 5 .5 6
* C C C C U C S C C C C S 4 5 C C S C C C S C .5 6


M ethodology ..............................58
Discussion of Results . . . . . . . .59
Sensitivity Analysis . . . . . . . 62
Some Informal Evidence . . .. . . . .65


67


APPENDIX A


APPENDIX B


PROPERTIES OF PORTFOLIOS


TESTABLE HYPOTHESES: AN ALGEBRAIC


ILLUSTRATION


. . . 85


REFERENCES


.88


BIOGRAPHICAL SKETCH


.93


Summary













Abstract of Dissertation Presented to the Graduate School of the University of Florida
in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy


VALUE OF LIQUIDITY IN FINANCIAL MARKETS

By

VINAY DATAR


APRIL


Chairman: Dr. Robert C.


Radcliffe


1994


Major Department: Finance, Insurance and Real Estate


In ideal markets, asset prices depend purely on fundamentals and

asset liquidity is not a concern because buyers (willing to pay the fundamental value)


can be readily found. However,


when markets are less than ideal, liquidity


considerations may affect equilibrium asset prices, reflecting the anticipated frictions

in trading. It is difficult to formalize liquidity because it may be influenced by a

variety of trading frictions related to various forms of market imperfections or


incompleteness.


Chapter


'tutorial'


on liquidity


It briefly reviews different types


of trading frictions that may affect liquidity of traded assets. Several, albeit noisy,

measures of liquidity are outlined. A simple model of liquidity is presented to

demonstrate the basic economics of liquidity. If liquidity has value then investors may








support for such liquidity premia in asset returns? Chapter 2 examines this question

and presents evidence of liquidity premia in the cross-section of stock returns. It is

suggested that the well known relationship between firm-size and returns may, in fact,

be due to liquidity considerations. Further, the relationship between returns and

liquidity is found to be robust to any potential seasonality in stock returns (i.e., the

'January effect'). Chapter 3 examines the impact of liquidity on the pricing of Closed-

End funds. Funds can potentially create value by buying illiquid assets and selling


more liquid claims.


The claims issued by such funds would be more valuable than the


underlying assets and would, therefore, trade at premium relative to the underlying


assets.


The empirical evidence is consistent with this notion. Specifically, in a cross-


sectional study, premia increase (or discounts decrease) as liquidity of a fund


increases (as measured by proxies).


Overall, the results suggest that liquidity


considerations may play an important role in the determination asset prices in

financial markets, and may plausibly explain some of the well-known anomalies in

financial markets.













CHAPTER


A TUTORIAL ON LIQUIDITY


Introduction


The notion of asset liquidity lacks a precise formal definition despite its


intuitive appeal.


Grossman and Miller (1988) express this idea quite succinctly:


Keynes once observed that while most of us could surely


agree that Queen


Victoria was a happier woman but a


less successful monarch than Queen Elizabeth I, v
would be hard put to restate that notion in precise


mathematical terms.
with equal force to ti
617).


Keynes's observation could apply
he notion of market liquidity. (page


Lippman and McCall (1986) observe that academic economists do not have a

definition of liquidity as a measurable concept, although there is a general agreement


that liquidity is the 'marketability'


of an asset; or the property of an asset that


facilitates immediate exchange for cash,


without affecting the market price.


They


define liquidity in terms of the time required to sell an asset to the best bidder (on


average).


Grossman and Miller (1988) suggest that liquidity is related to the 'price


concession' that may be demanded by a potential buyer to participate in an immediate

trade. Although a precise metric of liquidity remains elusive in theory, in practice








2

Stock Exchange) fact book (1990), the dollar volume of trade on U.S exchanges


exceeds $ 2 trillion per year, where the total market value of (potentially tradable)

listed assets is about $ 3 trillion. This amounts to an yearly turnover rate of about


70%.


This suggests that investors do care about the ability to trade, and are willing to


pay the attendant transaction costs. Further, the chaotic frenzy on the trading floors


suggests a strong sense of urgency to trade.


Why might so many traders fall over


each other (often times literally), to frantically engage in trade? How might trading,

or frictions in trading, affect long run prices? These questions form the central theme

of this study; what is the meaning and value of liquidity?

Liquidity considerations play no role in traditional asset pricing models; any


amount of trade can occur in equilibrium,


without affecting prices.


This result is


natural when markets are perfect. Here, the market value of liquidity is zero, because

market participants can supply liquidity at zero cost, if and when liquidity is

demanded. Recent literature on liquidity attempts to provide some insights into the


impact of trading (and frictions in trading) on asset values.


The general suggestion is


that less liquid assets have higher expected returns (or lower prices) than more liquid

assets.1 These higher returns, or liquidity premia, are a compensation demanded by

market participants to offset anticipated costs of trading.

Liquidity considerations provide valuable insights into asset pricing to the

extent that they explain, albeit in part, some well known empirical anomalies:








3
1) Claims to identical cash flows may have different prices [ See, e.g., Amihud and

Mendelson (1991), Boudoukh & Whitelaw (1991)]. 2) Small firms have higher

average returns than large firms [e.g., Stoll and Whaley (1983)]. 3) Closed-end funds

trade at discounts or premia relative to the market value of underlying assets [See

Chapter 3].


This chapter proceeds as follows:


The first section describes various motives


for trade, sources of potential frictions and prevalent measures of liquidity.


second section presents a simple model that captures the basic economics of liquidity


premia.


The third section concludes.


What Is Liquidity?: Overview of Literature


It is helpful to think about liquidity and its value in terms of the supply and

demand for liquidity. In particular, let us examine why people might want to trade

(demand liquidity); and why it might be costly to supply liquidity. An examination of

these aspects might suggest a clue as to the meaning and value of liquidity.


Why Do People Demand Liquidity?


Fundamentally, individuals would want to trade (i.e. demand liquidity) when

private valuation for an extra unit of an asset is different from market valuation. In

such situations there are gains to be made from trading, and this suggests a motive for








4

Literature has postulated several reasons for such a differential valuation,


where private value may be different from market value.


Unforeseen shocks to


preferences, endowments, information (about asset quality) and life-cycle trading may

lead to personalized values that are different from market value. For example,

Amihud and Mendelson (1986), Diamond and Dybvig (1983) and Flannery (1991)

consider a situation where individuals are hit by a preference shock; here some agents


early in the sense that such agents care only about current consumption, and they


have no value for future consumption. Clearly, these agents would be willing to sell

(to the highest bidder) at any non-negative price. In similar spirit, Campbell,

Grossman and Wang (1992) introduce change in the degree of risk aversion as a


motive for trade; here,


personal values are affected because of the change in risk


aversion and agents can gain by trading. Admati and Pfleiderer (1988), Glosten and

Milgrom (1985) and Kyle (1985) present models with private information as a motive

for trade; smart traders have private information about the true value of the asset, and

such traders can benefit by trading, as long as the market value is different from the

true value. DeLong, Shleifer, Summers and Waldman examine the impact of 'noise';

irrational traders perceive the market value to be too low or too high, and such


traders arrive in the market as buyers or sellers.


Constantinides (1986) and Grossman


and Miller (1988) introduce rebalancing as a motive for trade; individuals trade to

allocate optimal amounts of their wealth between risky and risk-free assets. In such








5
individual would hold; trading occurs until, in equilibrium, the private values of all

agents are equal to the market value.

In summary, the liquidity literature considers four main motives for trade:

1) private information about quality of assets (informed traders), 2) private beliefs

about quality of assets (noise traders), 3) need for rebalancing to maintain optimal

level of risky assets in the portfolio (liquidity traders) and 4) consumption needs

(liquidity traders).


Why is it Costly to Supply Liquidity?


When markets are perfect and complete, trading can occur at zero cost, and as


a result liquidity can be supplied at zero cost.


should be zero.


Therefore, the market value of liquidity


The literature examines the issue of the cost of liquidity in two


separate but related ways:


1) incomplete markets and 2) imperfect markets. In both


cases, there are costs associated with trading, and investors demand compensation to

offset the anticipated costs of trading. An increase in such anticipated costs, results in

a lower ex-ante price (or higher expected returns) for a given asset (or a claim to a


given set of cash flows).


The extra return due to liquidity considerations is called


liquidity premium.


Incomplete markets.


Let's consider the incomplete market framework.


When


markets are incomplete, state-contingent claims are not available for some states of










incomplete. Such an approach allows us to study the


'insurance'


aspect of liquidity


and the value of such an insurance.

Consider the following; investors may be driven to sell (demand liquidity)

because of an endowment shock, or because of an exposure to some other risk that is

not insurable, and all investors are exposed to such a non-diversifiable risk. In such a

situation, the investors may need to worry about an additional risk (in addition to

fundamental risk) because the expected resale price may be sensitive to trading (i.e.


doesn't depend purely on the asset fundamentals). For example,


Campbell, Grossman


and Wang (1992) introduce a personal shock to risk preferences as a non-diversifiable

risk. Such shocks increase the volatility of prices. Risk averse rational traders would

demand extra premium in such a market because they are not immune to the same


potential shock in the future.


This combination of non-diversifiable excess volatility


and risk-aversion gives rise to liquidity premia.2

Diamond and Dybvig (1983) show that illiquid assets have higher returns than

liquid bank deposits because, in effect, the liquid deposits provide guaranteed

consumption.3 In their model, productive investments are not reversible, and such

investments have only one known terminal payoff with no intermediate dividends.

Further, the claims on these investments are not tradable at intermediate dates (real


results


model


e ra s i m i l ar to


noise


trader


-- .-.-- ----- .- - --.-. - ,- -


model


presented


V








7
investments are illiquid). All individuals in the economy are identical (on date 0), but


a few of them (randomly) may 'die'


early (before the terminal payoff is received);


'diers'


need to consume early, therefore they would demand liquidity.


On date 0,


all individual know that they have some probability of being a demander of liquidity.

Investors may pass up the illiquid investment, despite the high returns that it provides.

A bank creates liquidity by issuing demand deposits and using some of the proceeds

to invest in the illiquid productive asset. Investors are willing to accept a lower return

on the liquid deposits (as compared to higher return on the illiquid productive asset).4

In summary, the mere possibility of any uninsured shock to beliefs, risk-

preference, time-preference, income, endowment, consumption liabilities or human

capital is can potentially, give rise to liquidity premia.5


Imperfect markets.


The liquidity literature considers a variety of market


imperfections that may generate liquidity premia.


Constantinides (1986) examines the


impact of transaction costs on expected returns. In his framework, investors trade-off

the benefits of optimal rebalancing against the cost (due to brokerage costs) of


trading; however he finds that the transaction costs have only a


'second order'


effect


on the risk premia. Intuitively, his results suggest that the need to rebalance may not

be an important reason to demand liquidity; as a result, investors can tolerate any loss


4 Jacklin (1987) points out that the liquidity premia in such a model, with a risk
neutral framework, crucially depend on market incompleteness (introduced here bv the








8
in utility due to sub-optimally balanced portfolios and forego liquidity rather than pay

transaction costs. Amihud and Mendelson (1986) suggest that investors do consider

the present value of anticipated transaction costs at the time of initial purchase, this

results in lower prices (or higher expected returns) for less liquid assets.


Market micro-structure.


The emerging literature on market micro-structure


considers the details of the market mechanism that facilitates trade, and examines the


attendant frictions in the trading process.


In such models, the traders arrive at


different (asynchronous) times, and a market maker stands ready to be a counterpart

for trades. Such a view (of the market) enriches the traditional view of the market as


a giant trading hall,


where symmetrically informed traders submit their demand


schedules and trade occurs, if at all, after the equilibrium price is determined.


market maker buys at the bid price and sells at the ask price, from any willing trader.

The difference between ask and bid provides fair compensation to the market maker,

to offset the costs of making a market.


The costs of market making may be broken down into three parts as


1) adverse selection costs [Glosten and Milgrom (1985)], 2) inventory management

costs [Ho and Stoll (1981), Grossman and Miller (1988)], and 3) order processing


costs [Roll(198,

market maker,


4)].


The adverse selection costs refer to the losses incurred by the


while trading with traders who have private information. In the


process of providing immediacy, the market maker may be called upon to allocate a









These costs are called the inventory management costs.


Order processing costs are


simply the administrative costs of maintaining the trading establishment.

The bid-ask spread, that potentially offsets the costs of making a market,

represents a friction in the trading process and is therefore viewed as related to


liquidity.


The quoted bid-ask spread is valid for a defined size of trade (usually


around 500 shares) and larger orders may temporarily change market prices.


of trade that may temporarily move prices by one dollar is called 'depth'


The size


and the


speed with the price


'recovers' to some long run


'natural' level is called the


'resiliency'.6 The bid-ask spread, depth and resiliency collectively characterize

market liquidity.


Different Measures of Liquidity


It is difficult to define a single metric for liquidity because liquidity is

influenced by several different sources of trading frictions. Several measures of

liquidity have been discussed in literature, some of which are reviewed below.


The bid-ask spread has been suggested as a measure of liquidity by


several


authors because it is a mechanism that compensates the market-maker who is

the provider of last resort for liquidity [e.g., Demsetz (1968), Glosten and


Milgrom (1985), Stoll (1989) and Amihud and Mendelson (1986)].


The market


maker may simultaneously trade at the bid (for a buy transaction) and at the










ask (for a sell transaction),


while profiting from the spread.


In a competitive


market for market-making services, the expected profits to the market-maker


would be a fair measure of the cost of providing liquidity services.


There are


some problems with this argument. Lee, Mucklow and Ready (1993) argue

that bid-ask spread, by itself, may be a misleading measure of liquidity


because the quoted bid-ask spread is valid only for a limited trade size.


They


suggest that depth and bid-ask spread, together, provide a better measure of


liquidity. In similar spirit,


Grossman and Miller (1988) argue that bid-ask


spread fails to measure the cost of immediacy (or the cost of delaying a trade);

and further, the bid-ask spread may not be a fair measure of market maker's

compensation, to the extent that the market maker may not simultaneously

trade at both the bid price as well the ask price. Several authors have noted

that the quoted bid-ask spread may not be the effective spread, because many

trades may occur inside the bid-ask spread, and spread may be an extremely

noisy (to the point of being meaningless) measure of liquidity.

Liquidity ratio, defined as average dollar volume of trade, per unit price

change over some interval, is sometimes used as a measure of liquidity [e.g.,


Copper,


Groth and Avera (1985)]. Grossman and Miller (1988) point out that


this measure fails to capture the impact of a larger than average trade, and

further it does not account for fundamental volatility. For example, release of










in this case, because the change in price is not caused by liquidity


considerations, but purely by fundamental re-valuation.


The more efficient the


market, the smaller would be the measure of liquidity ratio, thereby it would

understate the true liquidity.

Volatility ratio, defined as a ratio of long term return volatility to short term


return volatility, is suggested by Barnea (1974) as a proxy for liquidity.


This


ratio attempts to capture the price volatility caused by the order imbalance,

albeit imperfectly. A change in fundamental volatility can introduce noise to

this measure, because it cannot be distinguished from liquidity related


volatility.


This ratio may be a reasonable measure of liquidity to the extent that


fundamental volatility can be assumed to be stationary.

Auto-correlation of returns may serve as a proxy for liquidity. Grossman and

Miller (1988) and Goldman and Beja (1979) show that the serial correlation of

returns reflects the degree of participation of the market maker in the trading


process.


Highly traded securities should have a lower (closer to zero) serial


correlation of returns.


Thinly traded securities may have a larger (away from


zero) negative serial correlation of returns.


The price smoothing function of


the market maker may create some positive serial correlation, and the

inventory management process may introduce a negative serial correlation.

These opposing effects add noise to the measure but, a larger negative serial








12
Volume of trade is suggested as an indicator of liquidity some authors [e.g.,


Benston and Hagerman (1974); Stoll (1978); Barclay and Smith (1988)].


intuition is that high trading volume may mitigate the problem of inventory

management for the market maker, and thereby reduce trading costs; this can

occur if buy and sell orders arrive rapidly with equal probability, and the


market maker can simply cross orders.


Further, higher volume allows the


fixed cost component of administrative costs to be spread over a larger number


of shares, and the per share cost of trading is reduced.


Higher volume of trade


may indicate higher liquidity.

The next section demonstrates the relationship between trading frictions and the value

of liquidity, using a simple model.


Value of Liquidity: A Basic Model


In this section,


we develop a simple model that provides a framework to study


the basic economics of liquidity. An important contribution of this model is the

demonstration that liquidity premia can arise in a fairly simple framework and that a

richer framework, although enriching, is not essential for liquidity premia to exist.

The simplicity of the model helps identify the bare essentials of liquidity and its


relationship with long run value.


We suggest that liquidity premia may arise even in a


simple economy with dividend certainty, risk neutrality and ex-ante homogeneity










above assumptions (as modelled in related literature) creates a broader potential role

for trading and may, therefore, further increase the equilibrium value of liquidity.


The Economy


Consider an exchange economy with only two assets: a


'liquid' asset and an


'illiquid' asset; both the assets represent traded claims to a perpetual stream of


guaranteed dividends of $D per period.


The amount and timing of dividend payout is


known (to everyone) with complete certainty and in this sense both the assets are

identical.


'liquid' asset is different from the


'liquid'


'illiquid' asset in the following sense:


claim provides a guarantee of a resale price, P1, at any time in the


future. Further, the supply of this asset is infinite and it cannot be sold short.7 For


example, consider a redeemable U.S.


Government consol that has a guaranteed


redemption value but this consol cannot be sold short. In contrast, the


resale price of


'illiquid' asset is not guaranteed.8


SThis restriction is necessary to impose illiquidity on the illiquid asset, otherwise
everyone will prefer to short sell a liquid asset instead of selling an illiquid asset. In
other words, allowing short sale of the liquid asset implicitly provides a costless way of
disposing off the illiquid asset. Consider, for example, that U.S. treasury bills are liquid,
but can not be sold short.










The investors are identical ex-ante and everybody is risk neutral.


The risk-free


discount rate is r%. A subset of investors may experience a shock to their time-


preference, at some future time.


The ex-ante probability,


ir, of this event is common


knowledge but the outcome is known only privately ex-post.


This risk is not insurable


because it cannot be observed publicly.


Price of the 'Liauid'


Asset


The 'liquid' asset guarantees dividends as well as resale price. The amounts as

well as the timing of cashflows from this asset are known with certainty. The risk


neutral value of this consol is simply the present value of expected future cashflows

discounted at the risk-free rate.


D D
L (l+r) (l+r)2


Where,


L = the guaranteed redemption value of the liquid asset

S= the dollar amount of the risk-free dividend paid periodically

= the risk-free discount rate


Price of the Illiauid Asset


r-W-t------1 -rI I 1 t -
















- P -f
-a


Where,


= the market clearing price of the illiquid asset at any time, t, in


the future.


The initial price of the illiquid asset; and it is equal to the


private valuation of individuals who, by chance, are not affected

by a time-preference shock, at any given time t in the future.


= some unspecified function of the illiquidity of the asset, at time


t. In some sense the variable,


, reflects the price concession


demanded by the buyers.

Tilde (-) denotes random variables and all the random variables above are

restricted to be non-negative. Further, let's assume that the distributions of all the

random variables are stationary through time.


Equation (2) says that the resale price,


price, Po, net of any trading friction,


is equal to the buyer's reservation


The buyer's reservation price, in turn,


mlr /r i


1 ,I .- .- ^ ^ L J --- ^ ^1 t .li 1 -




















E( P M)


E( M)


The present value of these payoff possibilities is equal to Po,,


-7r) (D+Po)


(1+r)


(1 -i) (D+Po)


where


7 [D+E(PM)]


(1+r)


7 [D+Po-E(f)]


(1 +r)


(1+r)


D
r


Tr E(f')


_ ir E(f)
r


Equation (3) says that the initial price of an illiquid asset is lower than the

price of an identical liquid asset by an amount equal to the present value of expected


M fl an .-A A -I, *1,1 ... ZJ1 L_. _. .: 1 .5 5


D+


D+


D +


D +


1 1 1 1









a liquid asset, by virtue of a guaranteed resale price; such an option is not available

on an illiquid asset and the relative prices properly account for this aspect.

In our story the buyer's reservation price, Po, is stationary through time

(because the future prospects are unchanged); and it is not the same as the resale


price,


, (or the market clearing price) in the future.


Under these conditions, there


is further gain from trading and this is counter to the notion of an equilibrium. In


order to sustain these divergent prices in equilibrium,


we need some friction to


restrict further trading.9 It is this friction that captures the essence of the cost of

liquidity at the margin. For example, it can be seen from equation (3) that in a

frictionless world, E(j)=0, the price of an illiquid asset is the same as the price of a


liquid asset.


This makes sense because, by assumption, liquidity is freely supplied.


We discuss the interpretation of this friction,


later on.


We demonstrated that liquidity premia, higher returns10 or lower price, may

arise even in a simple economy with ex-ante homogeneity, risk neutrality and

dividend certainty. All that is needed to justify liquidity premia is some friction in the


trading process (and some motive to trade).


In essence, market


imperfection/incompleteness generates liquidity premia.








18

In our framework, investors' trading motive was generated by an uninsurable

shock to the time-preference, and the trading friction was in the form of sellers


receiving a price below the buyer's


reservation price.


Discussion:


Let's consider extreme values of the trading friction,


, to develop some feel


for its impact on prices. At one extreme, if we eliminate the trading friction, the


liquidity premia disappear. This can be seen from equation (3) by setting the value of

E(f) equal to zero. In other words, without trading frictions, all traded assets would


be fully liquid and their prices would be identical, for a given stream of dividends. At

the other extreme, if the trading friction were to be very large then the market would

not open and the resale price of the asset would be zero (the smallest possible price).

From equation (3) we have:


D
(r+7r)


This is analogous to a complete market failure, at the time of 'death',


probability of 'death' and the liquidation value is zero.


where 7r is the


We would expect the trading


friction to be in between these two extremes and this suggests a range for the value of

liquidity.


w I I I -










unchanged.


The market clearing price,


may change from time to time depending


on the trading friction,


', that is prevalent in the market at a given time.


proceeds of sale received by the seller,


are exactly equal to the buyer's


reservation price net of the trading friction, in other words the seller bears all the

costs associated with selling (and the ex-ante estimate of this cost is reflected in the

buyer's reservation price, Po, because at some time in the future the buyer may


become a seller).


We interpret the friction,


, as the per-share cost of operating the


market at any given time. If the cost of operating the market is fixed then the per-

share cost would be lower for an asset with a higher trading volume, because the


fixed cost is spread out over a larger number of traded shares, at any given time.


suggests that across different assets, the market price,


This


would be increasing the


higher the volume of trade. Alternatively stated, the observed returns,


would be decreasing the higher the trading volume. Further, this negative relation

between returns and volume gets stronger if we look at returns over longer holding


periods (say over n periods); because the random resale price,


becomes less


important and the dividends (n times D) play a larger role in the observed returns,


(PR +nD)/P


- 'WI


This result has an easy intuitive interpretation; investors can get higher


+D)/ ,








20

The essence of value of liquidity can be easily described in the context of the

model above. Investors know that they may have a liquidity need in the future, and

they consider the expected future resale price at the time of initial purchase. If the

expected future resale price depends purely on fundamentals then initial purchase

price also depends purely on fundamentals; and the market value of liquidity is zero.

However, if the future resale price is expected to be less than the fundamental price

then initial purchase price is reduced to that extent; and in equilibrium the deviation

from the fundamental value reflects the impact of liquidity considerations. Liquidity


considerations become important when there is a


'wedge' between the expected


proceeds to the seller and the reservation price of the buyer. An intuitive


interpretation of such a


'wedge'


may be as follows: when firms are liquidated, the


liquidation proceeds may be lower than the going concern value.


The difference


between the going concern value and the liquidation proceeds is an example of a


'wedge'


In financial markets, such a wedge can be caused by a variety of frictions to


trade; in our model such a wedge (t) was exogenous. Explicit considerations of the


details of the market mechanism, asset characteristics and the nature of the economy

may give rise to this wedge (f)." For example, brokerage costs, information


asymmetry


limitation on risk bearing capacity etc. can potentially create frictions


(i.e. the


'wedge'


or 'f') that result in liquidity being costly to supply


and therefore valuable.








21


Summary


The meaning and value of liquidity can be understood by looking at the


demand for and supply of liquidity; namely


why do people trade? and why it might


be costly to facilitate trade.

Literature suggests four main motives for why people trade (or demand


liquidity):


1) private information about asset quality (information motive), 2) private


beliefs about asset quality (noise motive), 3) need for rebalancing to maintain an

optimal level of risk in the portfolio (rebalancing motive) and 4) life cycle needs or

cash needs to meet consumption liabilities (consumption motive).

On the supply side, it may be costly to supply liquidity (facilitate trade)


because of three main reasons:


processing costs.


1) adverse selection, 2) inventory costs and 3) order


The model in the second section demonstrates, that the cost of


supplying liquidity creates a


'wedge' between the reservation price of the buyer and


the proceeds of sale received by the seller.


This


'wedge'


is the essence of the market


value of liquidity.


An explicit consideration of liquidity (the market value of liquidity), affects


ante prices of traded assets; i.e. prices depend on fundamentals as well as on the cost


of liquidity


For a given set of fundamentals, assets that are more costly to trade (less


liquid assets) are worth less; this reflects the market value of liquidity


Alternatively








22

over the fundamental returns (premium over hypothetical returns in absence of

liquidity considerations) is called the liquidity premium.

In essence, trading frictions impair liquidity; and liquidity premia arise as a

compensation for anticipated frictions in trading. Liquidity may reflect a composite

influence of several types of trading frictions, and therefore it is difficult to devise a


single metric for liquidity.


The bid-ask spread,


volatility ratio, market depth, auto-


correlation of returns and volume of trade provide a way of quantifying liquidity of a

traded asset.

Liquidity considerations may be the bridge between the perfect world of theory


and the clumsy imperfect world in practice.


Liquidity has important implications that


potentially explain, albeit in part, some of the well known anomalies in financial


markets.


The next two essays empirically examine the impact of liquidity premia on


traded assets. The first essay examines all the stocks traded on the NYSE and AMEX

stock exchanges. The evidence suggests that the cross-sectional variation of returns


may be related to liquidity.


The second essay suggests that the premia and discounts


in closed-end funds may be influenced by liquidity considerations.













CHAPTER


CROSS-SECTION OF STOCK RETURNS REVISITED:
LIQUIDITY PREM1A AND ROLE OF SIZE


Introduction


The well known relationship between size and returns achieved greater


prominence in a recent article by Fama and French (FF) (1992).

size plays a major role in describing the cross-section of returns,


completely fails to explain any variation in returns.


They suggest that

whereas CAPM just


The economic interpretation of the


role of size is an open issue. FF (1992) do not make any claims as to the underlying

economics, but provide a conjecture that size may capture some risk. In similar spirit,

Berk (1992) and Berk and Takezawa (1993) suggest that market value, to the

exclusion of physical size attributes (operating measures like sales, number of

employees, etc.), captures risk and therefore should be related to returns.


The notion that size captures some risk is appealing at first glance.


as shown by Keim (1983) and further confirmed by this study


relationship is not observed in non-January months.


However,


the size-return


The notion that size captures


some unobservable risk is not very useful if size is not even related to returns most of

the time (or in non-January months).








24

Literature on liquidity' suggests that impediments to liquidity may result in

higher equilibrium returns for illiquid assets (i.e. liquidity premia).? In this spirit,

James and Edmister (1983) offer the insight that the observed relationship between

size and returns may, in fact, reflect a more fundamental relationship between

liquidity and returns; however, in their data they do not find any support for their


conjecture.


In this study, using a broader data set,


we explore this conjecture and find


support for the notion that size may, in fact, reflect liquidity.

The main objective of this study is to investigate any influence of liquidity on


stock returns.


We find that the empirical results strongly support the notion of


liquidity premia. Specifically, thinly traded (low trading volume) stocks provide

higher average returns than highly traded (high trading volume) stocks. In order to


clarify the role of size vis-a-vis volume,


size and volume.


we examine portfolios sorted on the basis of


The relationship between volume and returns persists even after


controlling for size. Further, upon controlling for volume, the influence of size on

returns is insignificant. Moreover, in non-January months, size has no relationship


example,


Amihud


Mendelson


(1986)


Bhardwaj


Brooks


(1992),


Boudoukh and Whitelaw (1993), Constantinides (1986),


Demsetz (1968)


Diamond and


Verrecchia (1991), Flannery (1991),


Grossman and Miller (1988), Jacklin (1987), Pagano


(1989), Reinganum (1990), Shleifer and Vishny (1992), Stoll and Whaley (1983), Tinic
(1972) examine various impediments to trade and suggest a role for liquidity in asset


returns.


This a partial list and certainly not exhaustive.








25

with returns but volume continues to explain the cross-section of returns. Another

proxy for liquidity (the number of shares outstanding) shows essentially similar


results.


This suggests that, the relationship between size and returns may reflect a


more fundamental relationship between liquidity and returns.

We do not pretend to have explained the January effect. Size does have a very


strong relationship with returns and it is limited to January.


Our results suggest that


the observed overall relationship between size and returns seems to be subsumed by

liquidity proxies, and that the influence of liquidity is not limited to January.

In summary our findings suggest three important conclusions: (1) liquidity

premia do indeed exist in practice and they are pervasive (not limited to January), (2)

the observed overall relationship between size and returns may be due to liquidity


premia,


and (3) liquidity premia may play a major role in determination of relative


asset prices to the extent that size plays a major role per FF (1992).


In the first section,


we describe the data, our method of estimating beta and


proxies for liquidity. In the second section, using the same methodology as used by

FF, we examine the relationship between average returns and size as well as volume


and another liquidity proxy. We find that liquidity seems to have a stronger

relationship with returns than size does. This finding is further confirmed by the


Davidson and MacKinnon's J-test for non-nested hypotheses.


The third section


provides additional evidence in support of liquidity premia using test portfolios to








26

but the liquidity effect is not limited to January. In the fourth section, these findings

are further corroborated by the analysis of residuals using a methodology similar to


Chen (1983).


The final section summarizes and concludes.


Data and Methodology


Description of Data


The data set includes all firms on the NYSE exchange from the Center for

Research in Security Prices (CRSP). Monthly data is collected from the CRSP tape,


from July


1962 through December


The time period was selected to be


essentially similar to the time period selected by Fama and French (1992).

Log of firm size (LSZ.,_) is defined as the natural logarithm of total market


capitalization at the end of the prior month (or month t-l) for a given firm.


Log of


volume of trade (LVOL1..1) is defined as the natural logarithm of the average monthly

volume of shares traded in the prior 3 months (or month t-3 to month t-1).3 The log

of outstanding shares (LSH.1 ) is defined as the natural logarithm of the number of

shares outstanding at the end of the prior month (or month t-1).

Returns are expressed as a percentage change in the value of one dollar of


investment over the time period of interest.


We examine returns over three different


horizons; monthly, six-monthly and yearly.








27

Monthly returns are simply the returns in the current month (or month t).


Yearly returns are calculated as non-overlapping holding period returns.


These are the


returns for the current and the following eleven months; and reflect the percentage

change in value of one dollar invested over the time period. After the yearly return is

calculated for the current observation month (or month t), the next observation for the


firm is after


12 months (or month t+12). Six-monthly returns are calculated similar to


the non-overlapping yearly returns.

Using monthly returns does not provide an explicit role to the actual dividend


stream,


whereas using longer term returns does allow an explicit role to the


dividends.4 The cost of using long term returns is that we get fewer observations, and


implicitly we assume stationarity of the attributes under investigation.


Using


overlapping returns is one way around the problem of fewer observations, but this

may overstate the t-values to the extent the observations are not entirely independent,

we do not consider overlapping returns in this study.







4 It is important that the actual dividend stream be allowed to play an explicit role
in calculation of returns because we want to examine the long-run effect of liquidity on
a given stream of dividends. Prices may permanently deviate away from the 'fundamental
value' to compensate for illiquidity and this is the aspect that we want to investigate. For


example a consol of $


per period valued at $10 provides an implicit 10% return, and


the same console provides a 20% return if valued at $ 5. In reality, fundamental values









Estimation of Beta


Fama and MacBeth (1973) and Fama and French (1992) use the portfolio

approach to estimate betas in an attempt to reduce the noise in the estimate of betas.

Fama and French (1992) find that any beta measurement errors do not affect the

ordering of betas, and the portfolio approach is unnecessary for beta estimation.


Further, beta does not describe the cross-section of returns in their data. We expect

similar results because our sample and methodology are essentially similar. We do not


follow the portfolio approach, but instead, simply calculate betas for individual

stocks.


Since monthly returns are used, non-synchronous trading does not present a


major problem, so adjustment suggested by Dimson (1979) is not necessary.


values are calculated for each firm, in each month, using past 30 to 60 months of data

(based on availability) and using the value weighted portfolio of NYSE stocks from


the CRSP tape as a proxy for the market portfolio.


Proxies for Liquidity


The literature on liquidity recognizes that liquidity may have value as long as


markets are incomplete or imperfect in some sense.


Transaction costs, information


asymmetry or heterogeneity (of endowments, preferences or investment horizons) may










generate liquidity premia.6 However, there is no consensus about a good metric for


liquidity. For example, Amihud and Mendelson (1986)


use the bid-ask spread to


measure


the cost of immediate execution or the liquidity of an asset,


while Stoll


(1985) and Grossman and Miller (1988) argue against bid-ask spread as a measure of

liquidity (see, Chapter I), in spite of its rough common sense appeal as a metric for


liquidity.


They contend that, in general, the bid-ask spread fails to capture the market


makers' return for providing immediacy which is the cost to the investors for


receiving


immediacy.


In the context of alternative proxies for liquidity


Grossman and Miller (p.


630) argue that 'what we need is a measure of how well the market makers are

providing customers with an effective substitute for the delays in a search for a more


inclusive set of counterparties'.


They suggest that the ease with which one can find a


counterpart to a trade is an important attribute describing the liquidity of an asset.


In the spirit of Grossman and Miller,


we use the trading volume of a stock


(i.e., the average number of shares of a stock traded over the previous year), as a


measure of the liquidity of that stock.


The idea being, the higher the trading volume


of a stock, the easier it is to find a counterpart to trade with and hence, the greater


the liquidity of that stock and vice versa.


Higher trading volume, to the extent it


indicates that there are fewer impediments to trading, may be a good proxy for


liquidity.


We use monthly volume of trade, averaged over the prior three months, as










outstanding shares is a variable of interest because in Merton


's (1987) model, it


explains cross-sectional differences in stock returns. In fact, Merton (p.494) calls it


the degree of 'investor


recognition'


or the relative size of investor base. In general,


since the stocks having a larger investor base are also traded more frequently, the

number of shares outstanding can also be closely linked to the marketability or

liquidity of a stock.


Returns, Size and Liquidity


In this section we use the methodology used by FF (1992) and examine the


influence of size and liquidity proxies on the cross-section of returns.


The analysis


uses the basic methodology developed by Fama and MacBeth (FM) (1973) to test the

null hypothesis of no relationship between the explanatory variables and returns

(hypothesized dependent variable).


The FM regressions are carried out as follows: starting with January


1963,


perform a cross-sectional regression of returns on the explanatory variables. Such a


regression is carried out every month up to, and including, December


This


provides 348 monthly estimates of slopes for each of the explanatory variables.

Average slope is the arithmetic average across the time series of 348 monthly slope

estimates. Associated t-statistic is simply the average slope divided by its time series

standard error across the 348 monthly estimates of slope.7 The average slope in FM








31

tests represents the average effect of each explanatory variable on returns (the

hypothesized dependent variable).

Table I summarizes the results of monthly FM regressions. Size has an

average slope of -0.14 and the associated t-value is -2.58, this is about the same as


reported by FF (1992). Liquidity proxies,


volume and number of outstanding shares,


seem to show a somewhat stronger statistical relationship with returns than size does.


For example,


volume has an average slope of -0.12,


with a t-value of -2.95. LSH


(number of outstanding shares) show an average slope of -0.16,


with a t-value of


3.13. However, the explanatory power of size as well as liquidity is not significant


when these variables are used simultaneously.


This may be because of multi-


collinearity in the data and the attendant lack of precision in the estimates. As shown

in Table II, size and liquidity variables are highly correlated and this may, potentially,

create the usual difficulties of interpretation in a multi-variate analysis. Later on, in


the third section,


we use test portfolios to study the independent effect of size and


liquidity


7(...continued)


stationary,


slope


esti mates


are assumed


to be


serially


independent.


alternative aggregation method (viz.


Fisher test)


, does not rely on stationarity, but does


assume time-series independence. The Fisher test aggregates 'p-values' for the time-series
of slope estimates, to develop a statistic that is distributed as Chi square. In our data, the
results of the Fisher test are in agreement with the results from the FM methodology.
This provides additional support to the conclusions drawn using the FM methodology.








32

Davidson and MacKinnon's (1981) J-test can help assess the relative strength

of trading volume and size in explaining the cross sectional differences in stock

returns. Intuitively, Davidson and MacKinnon's (1981) J-test for non-nested models


works as follows. Suppose,


Then,


we want to compare two non-nested hypotheses, HI with


we first find the predicted values of the dependent variable using (say)


Next,


we run HI


using these predicted values as an additional explanatory


variable and look for the significance of the coefficient on this additional variable.


The coefficient of the additional


variable will be significantly different from zero if


the initial predicting equation, i.e., H2 is valid.


We then repeat this procedure using


predicted values from HI as an additional explanatory variable in H2 and evaluate

significance of the coefficient.

In our case, HI states that size explains returns on stocks while H2 states that

trading volume explains the returns on stocks:


HI: Rit


i + b1u Ln (Size,,,) + ei,


H2: R ,


+ d1 Ln (Volume,i.) + h,,


Given the two non-nested hypotheses,


First,


we carry out the J-test as follows.


we regress stock returns on trading volume according to H2 and find the


predicted values for returns.


Then,


we run H


using these predicted values of returns










H2 is 1.71 (t-value 2.13).


Given that the coefficient on the additional variable


(namely, returns predicted using H2) is significant, this half of the test provides

support in favor of H2, i.e., trading volume explains stock returns.


The second part of the test proceeds in reverse order. Namely, w<

returns on size according to H 1 and find the predicted values for returns.


e first regress


Next,


regress returns on these predicted values as an explanatory variable in addition to


trading volume.


We find that the average slope coefficient on trading volume is -0.80


(t-value -0.72) while that on the additional variable is


1.05 (t-value


.60).


Since the


coefficient on the additional variable (namely, returns predicted using HI) is not


significant, the conclusion is that the predicted values from H


do not explain the


returns significantly.


Therefore,


favor of


we conclude that Davidson and MacKinnon's J-test rejects size in


trading volume (or liquidity) as the explanatory variable of the cross section


of stock returns.


Size or Volume?, Evidence from


Test Portfolios


The high cross-sectional correlation between size and volume presents a

problem in that the estimates lack precision in presence of multi-collinearity. In this

section we follow the methodology used, in a similar but different context, by

Jegadeesh (1992); and construct test portfolios that have a low cross-sectional










returns. In contrast,

controlling for size.


volume has a strong negative relationship with returns even after

Volume absorbs the role of size in explaining returns and this is


true even when we include January months in the test.

Test Portfolios


Four types of test portfolios are examined to study the influence of size vis-a-


vis volume on the cross-sectional returns.


These portfolios are called size based,


volume based, size-volume based and volume-size based portfolios.

The first type of test portfolios, called the 'size based' portfolios is


constructed as follows: Once every year in July, the securities listed on the New


York


Stock Exchange (NYSE) are ranked based on the market capitalization of equity in the


prior month (or June) and 20 size-based groups are formed. Portfolio


equally weighted


represents an


holding of the smallest 5 percent of the firms and portfolio 20


consists of an equally weighted portfolio of the largest 5 percent of the firms.


This


procedure is repeated once every year from


1963 to


1990. Appendix A, Panel A


summarizes the time series averages of properties of this type of portfolios.

The 'volume based' portfolios are formed in a similar manner except that the

ranking of firms is done on the basis of average monthly trading volume over the last


year (at least past six months).

Appendix A, Panel B. Table I,


The properties of these portfolios are summarized in


II shows that the average correlation between size and


urlnlimi in r thfaca nrnrtrf'nline ;e h;hl-h /(0kr.t (" QC t, A / CX,










formed as follows: every year in July,


we form


10 size based groups as before. Each


of these 10 size based groups is further partitioned into 3 volume based groups.


break-points for volume based ranks are found each year by ranking all the NYSE

listed securities into three equal groups on the basis of the average volume of trade


over the prior year.


This way, thirty 'size-volume' based portfolios are formed once


every year by using an equally weighted combination of member firms in each group.


For example, thirty 'size-volume' based portfolios for the year


formed as follows:


1963 are


1963 we rank all the NYSE listed firms into ten


groups on the basis of size at the end of the prior month (or June 30'

of these ten groups is further subdivided into 3 volume based groups.


1963). Each one

The breakpoints


for volume are calculated by ranking all the NYSE listed groups into 3 equal groups

on the basis of the average volume of trade over the prior year (beginning on July


'1962 and ending on June 30'


1963).


Thirty equally weighted portfolios are formed


using these size-volume based groups.

year up to and including July 1' 1990


This procedure is repeated in July of every


. Appendix A, Panel C shows the time series


averages of properties of these portfolios.


Finally,


we form a fourth type of portfolios called the 'volume-size'


portfolios. These are constructed by using a similar procedure as above except, now

we first form 10 groups based on volume and then divide each of these groups into 3


size-based groups. Appendix A, Panel D shows the properties of these portfolios.











Fama-MacBeth Rezressions


The relationship between returns and explanatory variables, size and volume,

is examined using the procedure developed by Fama and MacBeth (FM) (1973). A

similar procedure is used by Fama and French (FF) (1992). Lys and Sabino (1992)


point out that grouping based tests, of differences of means, may lack power.


We do


not use the usual means tests but instead use the regression approach that may have

more power per Lys and Sabino (1992).

Four sets of FM regressions are performed; one regression set for each type of


portfolios.


The FM regressions are carried out as follows: starting with July


1963,


calculate log of size (LSZ,,,) and log of average volume (LVOL1,.i) for each


portfolio. LSZ,,1, and LVOL,11 represent the


June).


respective values in the prior month (or


The CRSP returns (expressed as a percentage change in the value of unit


investment) are calculated over a period from July of year t through June of year


t+1.


The next observation is made in July of the following year (or year t+ 1).


This


provides a maximum of 28 observations per portfolio and the periods covered are


non-overlapping.


The returns are regressed cross-sectionally, each year on the


explanatory variables.


Table 1V


shows the time series average of slopes of cross-sectional yearly


regressions of portfolio returns on size and volume.


The average slopes should be








37

The slope coefficients on size or volume (individually) are significant in the


size portfolios and in the volume portfolios. For example, as shown in


Table IV-


Regression set 2 (volume based portfolios), the average slope coefficients (t-statistics)

on size and volume are -1.83(-2.51) and -1.61(-2.68) respectively in the univariate


regressions.


The high correlation between size and volume in these portfolios (0.85 to


0.95 from table III) makes it impossible to decide whether the returns are related to

size or to volume (because size and volume are very good proxies for each other).

Additional tests are helpful in isolating the effect of size and volume on


returns.


This is achieved by using the size-volume and volume-size portfolios; when


we use size-volume portfolios or volume-size portfolios,


we are able to reduce the


correlation between size and volume to about 0.35, as can be seen from table III.

These sets of portfolios with reduced correlation (between size and volume) allow us

to clarify the role of size vis-a-vis volume in explaining the variation in cross-


sectional returns.


Table IV


shows that the average slopes of FM regressions are


stronger for volume relative to size, this is true regardless of whether we use only

size (or volume) as an explanatory variable or use size and volume simultaneously.


For example, as shown in


Table VI-Regression set 3 (size-volume portfolios), the


average slope coefficients (t-values) on size and volume are -0.36(-0.37) and -1.60(-


2.03) respectively in the univariate regressions.


Results shown in


Table IV-Regression


set 4 (volume-size portfolios) are similar to the results in Regression set 3.


These tests










cross-sectional variation in returns.


Table V


shows essentially similar results using a


different proxy for liquidity (number of outstanding shares).

In conclusion, size (independent of proxies for liquidity) is not related to

cross-sectional returns; any observed relation between returns and size may be due to

the relation between size and liquidity; in contrast, there is a strong negative

relationship between returns and liquidity (independent of size).

Table VI shows that size has no relationship with returns in non-January


months but trading volume is negatively related to returns.


set 3 (table VI),


For example, in regression


volume has a slope of -1.92 with a t-value of -2.67 but size has no


explanatory power, size has a slope with a wrong sign and a t-value of 1.57


Table


VII shows essentially similar results using a different proxy for liquidity (number of

outstanding shares).

Overall, the evidence suggests that liquidity proxies (trading volume or number

of outstanding shares) may absorb the role of size in explaining the cross-section of


average returns.


Further, the liquidity effect is pervasive (not limited to January).


Analysis of residuals


In this section we use the methodology that was used by Chen (1983) to


examine the explanatory power of CAPM relative to APT.


The main idea is simple:


we want to first allow size to explain the returns and then examine the residuals to see










exercise by reversing the order and check


f size can explain anything that liquidity


may have missed.

The basic approach is to carry out FM type cross-sectional regressions as

follows:


Model


t Ln sizee. .) +


+ d, Ln (volume.,.)


Model


+ g, Ln (volume., ) +


using monthly data for returns,


size (m,).


+ m11, Ln (sizei,.1)


we have 348 estimates of slopes on volume (d) and


The time series average of these slopes and the associated standard errors


are shown in table VIII. Similar analysis is done using two different horizons for


returns, six-monthly and yearly. In addition the


entire analysis is repeated in non-


January months as well


At first glance,


this approach appears superfluous (see


footnote #8) in that the multiple regression in


Table


, using both size and volume,


seems do the same analysis.


However


, this approach


different from the multiple


regression analysis.


Here,


we hypothesize that we have two alternative models.


model uses size as an explanatory variable and the second mode


uses volume as an


explanatory variable.


The main objective, in this approach


adequacy of a single model; i.e.


we want to


is to examine the


see if one model leaves any information








40

In non-January months, size has no additional explanatory power after


allowing liquidity (volume) to explain the returns.


For example, in monthly analysis,


size has a slope of -0.04 and a t-value of -0.73. Six-monthly and yearly analysis


shows similar results.


Liquidity (volume) has significant explanatory power and the t-


values on liquidity are stronger than size, in the entire analysis.

As pointed out by Chen (1983), the results in this section can be misleading if

a variable is not significant, by itself as in table I, but attains significance in residual


analysis.


We do not have this type of a problem in our analysis.


Overall, the evidence corroborates the suggestion in prior sections that size has

no explanatory power in non-January months but liquidity does explain returns


throughout the year.


These findings are robust to returns measured over different


horizons (monthly, six-monthly or yearly).


Summary


In Chapter


than more liquid asse

premia? In this chapt


it was suggested that illiquid assets may provide higher returns

ts. Is there any empirical support to this notion of liquidity

er, we examine this question by analyzing the returns that are


realized by stocks listed on the New York and American Stock exchanges.


The results


support the notion of liquidity premia.

Our main result is that trading volume has a strong negative relationship with








41

control for the effect of size to distinguish between the role of size and trading


volume.


The relationship, between returns and trading volume, persists even after


controlling for the role of size.


This evidence strongly supports the notion of liquidity


premia in financial markets, to the extent that trading volume is a good proxy for

liquidity.


Further,


when we control for volume, the size does not explain any variation


in stock returns. Another liquidity proxy


like the number of outstanding shares, has a


similar influence on the relation between size and stock returns. Simply stated,

liquidity proxies absorb the role of size in explaining the cross-section of returns.

These results are robust to different measurement horizons (for example, monthly


returns, six-monthly or yearly returns).


Moreover, the relationship between liquidity


and returns is not limited to January whereas size-return relationship is limited to

January.

The evidence suggests that liquidity provides a better description of the cross-

section of returns than size does. Specifically, liquidity may absorb the role of size


and the liquidity effect is pervasive (not limited to January).


Moreover, to the extent


that size plays a major role in describing stock returns [Fama and French (1992)], it

seems that liquidity premia may have a major role in the determination of relative

asset prices.









42

TABLE


AVERAGE SLOPES OF MONTHLY
RETURNS ON BETA, SIZE, VOLU


SHARES OUTSTANDING: JANUARY


CROSS-SECTIONAL REGRESSIONS OF
ME OF TRADE AND THE NUMBER OF


963 TO DECEMBER


The associated t-statistics are in parentheses


Starting with January 1963, log of


size (LSZn, ,_


log of volume (LVOL1.,), log of shares


outstanding (LSHi,.) in the prior month and beta (using


last 60 to 30 months of data) are calculated for each


firm. These explanatory variables are matched with CRSP returns (expressed


as percentage change in value


of unit investment) over the current month. The next observation is made in the following month.


provides a maximum of 348 observations per firm.


Returns are regressed each month on the expj)lanatory variables.


Average


slopes are tlhe time


series


-, I


Beta LSZ LVOL LSH

0.06 (0.31)

-0.14 (-2.58)

-0. 12 (-2.95)

-0.16 (-3.13)

-0.04 (-0.21) -0.14 (-2.68)

-0.12 (-1.30) -0.02 (-0.25)

-.17 (-1.33) 0.05 (0.38)

-0.037 (-0.25) -0.24 (-1.81) -0.94 (-3.37) 0.88 (2.70)


I








43

TABLE II
AVERAGE CORRELATIONS BETWEEN SIZE
VOLUME OF TRADE AND THE NUMBER OF SHARES OUTSTANDING:


JANUARY


1963 TO DECEMBER


All correlations are significant at 0.01


(0.0001) level


Starting with January 1963, log of


size (LSZ


.,,I), log of volume (LVOL,.,.), log of shares


outstanding (LSH,,.) in the prior month are calculated for each firm. The next observation is made in the

following month. This provides a maximum of 348 observations per firm. The correlations are calculated

simply from the pooled time series and cross-sectional data.


LSZ LVOL


LVOL 0.72


LSH 0.88 0.84






a
*-2 44


,- 0 .

*u --
c "3a--: 4 --4

3o a -;f
0 Cu -^ 5 -"
2 t -. s VU -
c- -- ca"
*2 F 1
f C7,cc C J -'
F: .2 'S .2^^o .
'4) 13 ^^^
6^ -2 -" "i
0 Cso*
C ^S

5 cn Cu

'S C 'S -
>'' ft t4 fc. -%
-- 7 3> 'S '" .9 U


h aod^ -
-^ #-'' Cs-^* 'g
fT ; o- -o _- CU^ -'^S
L14 C.Q O ft.i


^ ____ S- --MDS .-
-^ 4- ti 3 <- 13
- 0 .4-' .
CU .- '-I ^- g*
C.' O5 -S Cu a^ ^'
cU 2?ES*C?>"
_n U 3- C 1


eII C0 ft3-
oo 5 ^:
4 .- C -^"c
0 C)A I.-f I. -o-
__ -- *0 2 c
CU U G) ft c 5
o 0 S *- Fu -
cu -^ .s C3 *
-S CU F
0 .') .9U -
*-. z 3 E = t -^
In -i S -" Cu
C) 3- 0 Cu Se






CU.- v 2 CU
C4-. 3- -
CU











TABLE


AVERAGE SLOPES OF FAMA-MACBETH (FM) REGRESSIONS, 6/63 TO 6/90


Regression Set 1

Size based Portfolios
20 portfolios per year for 28


years


Coefficient


of Size


-1.56 (-


-.77 (-0.50


h,. Coefficient of Volume


-3.02 (-2.14)
-1.03 (-0.54)


Regression Set


Volume based Portfolios


20 portfolios per year for


years


Coefficient of


-1.83 (-2


-2.09 (-0.9


b,, Coefficient of Volume


-1.61 (-2.68)


.27 (0


ression Set 3


Size-Volume Portfolios
30 portfolios per year for 28


years


b,, Coefficient of


-0.36 (-0


-0.24


(-0.22)


Coefficient of Volume


.60 (-


.24 (-1.28)


Regression Set 4
Volume-Size Portfolios


30 portfolios and


years


bh, Coefficient of


-1.05 (-


-0.90 (-0.81)


b,, Coefficient of Volume


(-2.07)
(-1.23)


The associated t-statistics are in parentheses


Four separate sets of FM regressions are carried out as follows: starting with July
calculate log of size (LSZ1.,) and log of average volume (LVOL,4,) for each portfolio. LS


1963, we


LVOL1., represent the


respective values in the prior month (or June).


The CRSP returns (expressed as


a percentage change in the value of unit investment) are calculated over a period from July of year t


through June of year t+ 1.


The next observation is made in July of the following year (or year t+ 1).


This provides a maximum of 28 observations per portfolio and the periods covered are non-


overlapping.


The returns are regressed cross-sectionally, each year on the explanatory variables.


The time series averages of slopes of cross-sectional yearly regressions of portfolio returns on


size and volume are presented below.


The t-values for average slopes are based on the time series


standard error of the estimates of mean slopes, t-values are shown in parentheses.
For each of the four regression sets, the basic cross-sectional regression is:









46

TABLE V
AVERAGE SLOPES OF FAMA-MACBETH (FM) REGRESSIONS, 6/63 TO 6/90


Regression Set 1

Size based Portfolios
20 portfolios per year for 28


years


b,, Coefficient of


-1.56 (-1.91)


-0.63 (-0.29


b,, Coefficient of LSH


-2.29 (-1.95)
-1.68 (-0.62)


Regression Set


LSH based Portfolios
20 portfolios per year for 28


years


Coefficient


of Size


-1.77 (-2


-2.04 (-0.74


b,, Coefficient of LSH


-2.15 (-2

0.37 (0.


session Set 3


Size-LSH Portfolios
30 portfolios per year for 2


years


bh, Coefficient


-0.88


-0.77


-0.47


b,, Coefficient of LSH


-1.82 (-2.98)
-0.83 (-0.54)


Regression Set 4
LSH-Size Portfolios


30 portfolios and


years


b,, Coefficient of


-1.41 (-


-0.88 (-0.48)


Coefficient of LSH


-2.29 (-3
-. (3


-1.63


The associated t-statistics are in parentheses


Four separate sets of FM regressions are carried out as follows:


starting


with July


963, we calculate log of size (LSZ;,,) and log of average LSH (LSH,,.) for each portfolio.


LSZi,,,


and LSH,1,.1 represent the


respective values in the prior month (or June).


The CRSP


returns (expressed as a percentage change in the value of unit investment) are calculated over a


period from July of year t through June of year t + 1.


The next observation is made in July of


the following year (or year t+ I). This provides a maximum of 28 observations per portfolio
and the periods covered are non-overlapping. The returns are regressed cross-sectionally, each


year on the explanatory variables.


The time series


averages


of slopes of cross-sectional


early regressions of portfolio


returns on size and LSH are presented below.


The t-values for average slopes are based on the


time series standard error of the estimates of mean slopes. t-values are shown in parentheses.


For each of the


four regression sets, the basic cross-sectional regression









47

TABLE VI
EXCLUDING JANUARY, AVERAGE SLOPES OF FM REGRESSIONS, 6/63 TO 6/90


Regression Set


Size based Portfolios
20 portfolios per year for 28


years


b,, Coefficient


-0.09


of Size


(-0.12)


(1.20)


b,, Coefficient of Volume


-0.41 (-0.34)


-2.55


Regression Set


Volume based Portfolios


20 portfolios per year for


Regression Set 3
Size-Volume Portfolios
30 portfolios per year for 28


years


years


Coefficient of


-1.09 (-1.83)


-0.49 (-0.24


h,, Coefficient of


b,, Coefficient of Volume


-0.96 (-1.94)
-0.47 (-0.27)


Coefficient of Volume


-1.92 (-2.67)
-2.12 (-2.38)


Regression Set 4
Volume-Size Portfolios


30 portfolios and


years


b,, Coefficient of Size


(0.90)


Coefficient of Volume


-1.99 (-3.06)


-1.99 (


-2.58)


The associated t-statistics are in parentheses.


Four separate sets of FM regressions are carried out


: starting with July


1963,


we calculate log


of size (LSZI,) and log of average volume (LVOL .,) for each portfolio. LSZAI


and LVOL,,,


represent the


respective values in the prior month (or June).


The CRSP returns (expressed as a


percentage change in the value of unit investment) are calculated over a period from July of year t


through June of year t+ 1; returns in January are ignored.


The next observation is made in July of the


following year


(or year t+1).


This provides a maximum of 28 observations per portfolio and the


periods covered are non-overlapping. The returns are regressed cross-sectionally, each year on the
explanatory variables.
The time series averages of slopes of cross-sectional yearly regressions of portfolio returns on


size and volume are presented below.


The t-values for average slopes are based on the time series


standard error of the estimates of mean slopes, t-values are shown in parentheses.
For each of the four regression sets, the basic cross-sectional regression is:









48

TABLE VII
EXCLUDING JANUARY, AVERAGE SLOPES OF FM REGRESSIONS, 6/63 TO 6/90


Regression Set


Size based Portfolios
20 portfolios per year for 28


years


b,, Coefficient of


-0.09 (-0.12)


(1.49)


b., Coefficient of LSH


-0.18 (-0.18)


-5.31 (-1


Regression Set


LSH based Portfolios
20 portfolios per year for 28 years


efficient of


-0.72


(-1.19)


0.79 (0.32)


b_, Coefficient of LSH


-0.87
-1.85


(-1.20)
(-0.65)


Regression Set


Size-LSH Portfolios
30 portfolios per year for 2


years


b,, Coefficient of


1.24(1


b,, Coefficient of LSH


-1.80 (-2.68)
-3.40 (-2.42)


Regression Set 4
LSH-Size Portfolios


30 portfolios and


years


b,, Coefficient of


0.61 (


0.60)


b Coefficient of LSH


-2.63 (
-4.10(


-4.11)
-2.94)


The associated t-statistics


are in parentheses


Four separate sets of FM regressions are carried out: starting with July 1963, we
calculate log of size (LSZ .,) and log of the number of shares outstanding (LSHi.i) for each


portfolio.


LSZ.,I and LSH 1., represent the


respective values in the prior month (or June).


CRSP returns (expressed as a percentage change in the value of unit investment) are calculated
over a period from July of year t through June of year t+ 1; returns in January are ignored.
The next observation is made in July of the following year (or year t+1). This provides a
maximum of 28 observations per portfolio and the periods covered are non-overlapping. The
returns are regressed cross-sectionally, each year on the explanatory variables.
The time series averages of slopes of cross-sectional yearly regressions of portfolio
returns on size and volume are presented below. The t-values for average slopes are based on
the time series standard error of the slope estimates, t-values are shown in parentheses.
For each of the four regression sets, the basic cross-sectional regression is:








49

TABLE VIII


RESIDUAL ANALYSIS


, AVERAGE SLOPES OF FM REGRESSIONS,


6/63 TO 6/90


Monthly


All No-Jan
Months

Regression Set -0.18 -0.09
1 (-4.34) (-2.65)

Volume, d


Six-Monthly


All No-Jan
months

-1.20 -0.63
(-4.45) (-3.09)


Yearly


All No-Jan
Months
-2.28 -1.41
(-3.59) (-2.58)


Regression Set
2


-0.04
(-0.73)


-0.20
(-0.56)


-0.62
(-0.77)


The associated t-statistics are in parentheses.


Using Chen's


(1983) methodology, returns are regressed cross-sectionally on


size (volume) and


the residuals from this regression are regressed on volume (size).


This analysis is carried out in six


different ways, using three different horizons for returns (monthly, six-monthly and yearly returns) and
using two different calendar periods for each horizon (including January and excluding January).


The time series averages of slopes of cross-sectional FM regressions are presented below.


slopes show the average effect of each explanatory variable on the residual returns.


The average


The t-values for


average slopes are based on the time series standard error of the estimates of mean slopes, t-values are
shown in parentheses.


For each of the


regression sets, the basic cross-sectional regression is:


Regression Set 1:R,1


= a, + b, Ln


(size, ) +


= c1 + d1 Ln (volume.-.) +


Regression Set


g, Ln volumeme~.


= h, + m, Ln (size,.,) +


TlrP ctmr.a c snfl eruoe ,Vcli rtn nnnn. .l\t u. II( n k.*ln. 41th' *r.













CHAPTER 3
IMPACT OF LIQUIDITY ON PREMIA/DISCOUNTS IN CLOSED-END FUNDS


Introduction


Closed-end funds are traded bundles of traded assets.


The market values of


these bundles are often different from the collective market values of their contents,

leading to observed premia or discounts. At first glance, these premia and discounts

appear irrational because they seem to suggest that claims to identical cash flows may

have different prices.

Several explanations of discounts have been proposed in extant studies'. Most

of these explanations motivate discounts through some special costs associated with


the bundling of assets.


However, these studies do not make any particular reference to


the form of the bundle (open-end or closed-end) or the type of bundled assets (equity

or bond). As a result most of the existing hypotheses are unable to explain the

following observations:


Premia and discounts may exist simultaneously across funds (Barclay et


1993).


Different types of funds (equity or bond) may have different


premia/discounts on average (Barclay et al.,1993).








51

Closed-end funds trade at zero discounts when they are open-ended (by

legal fiat)2.


Liquidity considerations, potentially


explain the above observations and thus


provide an appealing explanation of the premia/discount phenomenon.


The connection


between premia/discounts and liquidity is straightforward. Amihud and Mendelson

(1986) suggest that in equilibrium, the expected friction in the trading process is


reflected in current market value.


This implies that a bundle of assets is worth less


than its contents if it is less liquid than its contents In the context of closed-end

funds: discounts are observed when a fund is less liquid than the assets in its portfolio

and, premia are likely when a fund is more liquid than the holdings in its portfolio.

Subrahmanyam (1992) and Gorton and Pennacchi (1992) provide an

explanation for why the liquidity of a fund may be different from the liquidity of its


portfolio.


They suggest that security specific private information is diversified away in


a 'traded basket of securities'


(fund).


This information diversification has, potentially,


two conflicting effects on liquidity of a traded basket: first, the uninformed investors

face less information asymmetry (when trading a basket) and this may improve

liquidity. Second, the basket has lower trading activity (than the assets in the basket)


2 This is documented by Brickley and Schallheim (1985). Fund related costs that potentially
justify discounts (management fees, tax-timing loss etc.) are applicable to open-end funds as well.
The 'Cost' based explanations of discounts suggest that the market value of an open-end fund


overstates its true value.


This implies that, in equilibrium, the open-end funds should be fully


^- ^ I ^ > ^ ^ L - ^ - ^ ^ - > <








52

because the informed traders are driven away and the per share cost of trading may

go up to the extent that fixed costs of market making are spread over a smaller


trading base4.


The net impact of these two conflicting effects on liquidity may


increase or decrease the liquidity of a fund relative to its portfolio and this, in turn,

results in observed premia or discounts.

The objective of this chapter is to examine the relationship between liquidity


and premia/discounts.


We document the influence of liquidity on premia/discounts


using a variety of techniques including a single latent variable specification.


results strongly support the liquidity conjecture: funds with higher liquidity, as

measured by proxies for trading activity, have higher premia (or lower discounts) than


funds with lower liquidity.


This relationship is observed in equity funds as well as in


bond funds. Further, the results are robust to various assumptions about model

parameters and error structures.


The organization of this chapter is as follows: The first section develops

testable hypotheses relating liquidity to premia/discounts. The second section presents

data and methodology. Results are discussed in the third section. The fourth section


presents sensitivity analyses.


The fifth section shows informal evidence suggesting


that closed-end funds may behave like low-volume stocks.


The final section concludes.








53


Liquidity and Premia/Discounts


How is Liquidity Related to Premia/Discounts?


As discussed Chapter


1, the impact of liquidity on market value may be easily


stated, namely the market value of a traded asset is equal to the


'fundamental value'


minus the expected loss due to illiquidity. This simple notion may explain the

premia/discounts in closed-end funds. For the sake of exposition let's assume that a

closed-end fund holds a fixed portfolio of traded assets, although in reality the

portfolio may be dynamic. An investor can buy a claim (to a given set of cash flows)


directly in the market or indirectly through a fund.


The claim ownership through a


fund would be worth more/less than direct ownership if it (ownership through a fund)

reduces/increases expected costs of trading. For example, consider a traded security

(like I.B.M.) and suppose that one share of such a security represents a claim to cash


flows that would be worth $10 if there were no trading costs.


expected trading costs were $1


However, if the


(value of liquidity) then the market value of such a


share would be $9 ( or $10-$1). Now consider a closed-end fund that holds one such


share as the only asset in its 'portfolio'


and the fund issues one share.


The net asset


value (NAV) of the fund would be $9 (the market value of fund's assets).


of the fund represents a claim to cash flows that are worth $


One share


0 if the fund share can


hP tradC d x/ith 7,=rr tr..rlno r',tv ;ftL//r f"tlh trrl,'^n rnctc 1E, r, 4j) th1n thn,










market value of the fund share would be $8 (or $10-$2). In this situation the market


value of the fund, $8,


would be less than the NAV


of the fund, $9, and this results in


a discount of $1 on the fund share relative to the NAV


of the fund.


Conversely, if the


trading costs of the fund share were $0.5, then the market value of the fund would be


$9.50 and the fund would trade at a premium of $ 0.50 relative to its NAV


Notice


that the fund would be at a premium/discount depending on whether it is more liquid

or less liquid than its portfolio.6


Testable Hypotheses


Our central conjecture is that premia/discounts vary depending on the liquidity

of the fund shares relative to the liquidity of the assets held in the fund's portfolio. A

direct test of this notion is difficult to implement without any information about the

composition of a fund's portfolio. However, an indirect test can be devised using the

classical way of analyzing variability of an endogenous attribute by examining any


variation


'within-groups'


and 'across-groups'


This approach results in two hypotheses


described below:


Within a group of funds that hold similar assets (equity or bond),


premia increase (discounts decrease) as the liquidity of iunds


' shares


" Note that the open-end funds (OEFs) are traded in essentially a batch market.


Investors


place buy and sell orders directly with the fund and the fund can cross orders at the end of the
A9I ; \/at thi- nraf,>\









increases.


To see this, let us consider two closed-end funds that hold identical


assets but have different liquidity


. The NAVs of both the funds would be


equal to each other (by design) but the market values would be different

depending on liquidity. A fund that is more liquid would have a higher market

value resulting in a higher premium or lower discount than the other fund.

This is illustrated in more detail in Appendix B.

2) Average premia/discounts may be different across different types of

funds (equity versus bond). Let us consider two groups of funds, equity funds


and bond funds.


The average trading costs for equity assets are likely to be


different from bond assets because these costs relate to different types of assets


that are traded in different trading structures.


On the other hand, the average


trading costs for the shares of these funds are likely to be similar because they


both are equity securities that are traded in similar trading environments.


main distinction between the two groups of funds is in respect to the portfolio-

assets and differential liquidity of underlying portfolio-assets may result in

different average premia/discounts across different types of funds. Appendix B

provides a more detailed illustration of this idea.








56


Proxies for Liquidity


Literature on liquidity suggests that trading costs decrease in trading activity

and increase in the volatility of a traded asset.8 We use weekly volume of trade,

weekly dollar volume of trade and percentage of shares traded in a week as proxies

for trading activity. Normalized price range (maximum price minus minimum price

divided by minimum price) over a week is used as a proxy for volatility.


Merton (1987) proposes that liquidity increases in


asset.


'investor recognition' for the


This suggests that the number of outstanding shares may be a proxy for


liquidity. Stoll and Whaley (1983) suggest that trading costs decrease in firm size.

This suggests that market capitalization (size) may be a good measure of liquidity.

The next section describes the data and methodology that we use to examine

the hypotheses outlined in this section.


Data and Methodology


Data


Weekly data for funds listed on the NYSE is obtained from the CRSP (Center


for Research of Securities Prices) daily tapes.


Each week covers all the trading days


between Friday and Thursday.


19/1l/O1


The net asset value (NAV) data from


iQC5 opnrrnaic1t\ nrrnl\irl\f hlp\ flr lcormnnt A drli/icnrc


1/4/88 to


ThP N1AV










calculated upon close of trading on


the Wall Street Journal.


Thursday and are reported in the Friday issue of


The sample contains 18 domestic equity funds and 90 bond


funds.9


The natural logarithm of all raw variables is used to reduce the skewness in


the data; this way the outliers are


'pulled in'


and do not overwhelm the analysis. Log


of firm size (LSZ,) is defined as the natural logarithm of total market capitalization at

the end of week t, for a given fund i. Log of volume of trade (LVOLDi) is defined as

the natural logarithm of the volume of shares traded in week t. LDOL, is the log of


dollar volume of trade in week t, for fund i.


The log of outstanding shares (LSHH,) is


defined as the natural logarithm of the number of shares outstanding at the end of

week t. LTURN, is log of percentage of outstanding shares traded in week t.


Premium is measured as


prices


-NAV1i


NAVi,


X 100.


LPREM., is defined as


price,
Log( NV).
NAV,


LSIG1, is a measure of volatility defined as the log of normalized price range

(maximum trade price minus minimum trade price divided by minimum trade price)

for fund i over week t.








58


Methodology


We use the ordinary least square (OLS) method as well as the latent variable

method to examine the relationship between premia/discounts and proxies for


liquidity.


The OLS specification is:


LPREM,,


+ 3, (liquidity proxy),1,_l


where, i is an index for funds and t is an index for time.

The OLS method is known to be biased when explanatory variables are


measured with error.


In the context of liquidity, the proxies for liquidity are likely to


measure the true liquidity with considerable error; and OLS may be unreliable for the

purpose of estimation. In such situations, the latent variable approach has many

advantages over the OLS method and this is discussed next.


The Latent Variable specification treats liquidity as an

variable."~ Several measurable variables such as size (LSZ1,,),


'unobserved'

volume of trade


(LVOL1,t.), dollar volume of trade (LDOL1.[.) and turnover rate (LTURN,L.I) are

treated as error-prone proxies for liquidity and the attendant errors-in-variables are

explicitly specified.

The latent variable model is:


LPREM1,


= ^ ( 4)


+ Oi,,


= 21 (4i) + 02it


-= 1,










= -- 3 (ki)


+ 3it


= P4, (Pi) + 4


where,


Y3 and Y4... are some proxies that measure unobservable liquidity


(4) with error (0), the subscripts explain the time-series and cross-sectional


restrictions that are imposed on the parameters.

system if we have more proxies available for <.


We can add more equations to the

These equations are estimated


simultaneously as a system with f, 4 and 0 as parameters.


The econometric set-up


implies restrictions on the covariances (among observed variables), and these


restrictions can be expressed in terms of the parameters of the system.


Using this


underlying principle, the estimation procedure uses information in the sample

covariance structure to estimate parameter values." Such a technique provides

consistent estimates of parameters. Further, any omitted variables in the list of 'Y'

variables above, do not affect the consistency of parameter estimates.'2


Discussion of results


Table IX shows descriptive statistics of the sample.


The mean premia are


about -12%


for equity funds and about -1%


for bond funds. Several tests were carried


out to examine if these mean premia are statistically same or different, and all tests


indicate that the mean premia are different at 0.0


level. For example, the equality








60

of means test (t-test) for means from two independent random samples [See,


Greene (1990)] was carried out.


The null hypothesis that the mean premia are same


for bond funds and equity funds is rejected with a t-value of -55.64 (for the test

statistic) under assumption of unequal variances across samples; and under the

assumption equal variances, the t-value is -66.89. Such rejections may be misleading

if the underlying distributions (of means) are not approximately normal.'3 To ensure

that our results are not driven by the violation of distributional assumptions, non-


parametric tests [See,


Conover (1980)] were also carried out.


The Wilcoxon rank-sum


test rejects the hypothesis of equal premia (across groups of bond and equity funds)


with a Z score of -52.33,


which is significant at 0.01


The Kruskal-Wallis test


rejects the null with chi-square (1 degree of freedom) value of 2738.6,


value of 0.0001.


which has a p-


Thus, hypothesis #2 is supported in our sample; average premia in


equity funds are significantly different from average premia in bond funds.


Table X examines equity funds.


There is a significant cross-sectional


relationship between premia/discounts and various liquidity proxies.


For example,


estimates of slopes for all the variables are significant at


level.


This finding


supports Hypothesis #1


in the first section; premia increase as funds' liquidity


increases given that the underlying assets have similar liquidity. Literature suggests


that the cost of market making increase as volatility increases." Thus,


we would


* I I *x j j ,^ r j -* ^ .*^* -r.n *r\k t ..< ^ E... .L -- .-. _.I / .^ j










expect liquidity to decrease as volatility (LSIG) increases.


Table X shows that the


slope on volatility (LSIG) is negative, supporting the notion that premia are positively


related to liquidity.


Table Xl shows similar results for bond funds.


The ordinary least square method may be subject to errors-in-variables bias.

Such a bias attenuates the slope towards zero and the effect on t-value is ambiguous.

We use latent variable technique to correct for errors-in-variables later on. Further,


the OLS


method, as used in


Tables X and XI, may be biased to the extent log of


price may, indirectly, introduce a lagged dependent variable bias in small samples.


account for this potential bias,


we carried out cross-sectional regressions on a weekly


basis; and the result of this analysis was substantively similar.

Table XII shows the results of latent variable analysis using the data on equity


funds. Each row in the table represents a separate model.


We would expect positive


coefficients on liquidity for all variables except for LSIG. All models support the


notion that premia are increasing in liquidity (Hypothesis #1


in the first section). For


example, the slopes on liquidity (Ct) with premia (LPREM) are positive and significant


level.


The slopes on liquidity (4) for various proxies for liquidity, are in the


expected direction (except for LSIG in a few models) and significant at


level.


These results, taken together, suggest a positive relationship between premia and


liquidity. Bond funds show essentially similar behavior as shown in


Table V


, adding


further support to the liquidity conjecture.








62

significantly different across different types of funds. For example, discounts are


more likely in equity funds than in bond funds. This finding suggests that the

observed mean differences in premia (as shown in Table IX) are not driven by


outliers.


To examine the stability of findings,


different sub-periods.


we repeated the above analyses in several


The findings (not reported here) are substantively similar and


lend further credence to the above discussion.


Overall, in our data,


we find strong support for the notion that


premia/discounts in closed-end funds are related to proxies for liquidity of the claims

issued by the funds. In the next section we examine the robustness of these results to

various econometric specifications.


Sensitivity Analysis


The sensitivity of the results described above is examined with respect to

various assumptions about the intercepts, coefficients and the error structures.


Specifically,


we examine


1) panel data models with fixed and random effects in the


intercepts, 2) random coefficient model, and 3) Seemingly unrelated regression (SUR)


models. All of these methods are discussed in Greene (1990).


The results are


generally found to be robust to various specifications and thus provide additional

support to the findings described above.








63

This model assumes that differences across time periods (regarding average


premia and other industry wide effects) are reflected in the intercept.


The model is


written as:


LPREMI,


+ / (liquidity proxy)1,,


is an index for funds and t is an index for time.


The slopes on liquidity proxies using the FEM are reported in


coefficients of all the explanatory variables are significant at


Table XV


level and the signs of


the coefficients are exactly in line with the implications of the liquidity conjecture.

For example the turnover rate (LTURN) has a coefficient of 0.023 and a highly


significant t-value of 9.30.


The volatility measure (LSIG) is negatively related premia


with a coefficient of -0.036 with an associated t-value of -1


2) Panel Data Analysis: Random Effect Model (REM)

The REM specification allows the intercepts to be random across time in

contrast to the earlier FEM that treats the variation in intercepts as parametric shifts.

The REM is more general in that the selected time periods are treated as a sample


from a broader population.


The REM is written as:


LPREM,


where


j3 (liquidity proxy).t-l


is an index for funds, t is an index for time and u, is a mean-zero


random disturbance reflecting a random industry wide changes in the level of

average premia and other effects common to the industry.


=- Ot,


= +








64

3) Random Coefficients Model (RCM)

The RCM is even more general than the REM in that the coefficients,

including the intercept, are treated as random across time while the cross-correlation


and heteroscedasticity of the coefficients is taken into account.


The RCM specification


does not assume that the parameter estimates are independent drawings from a


stationary distribution.


Therefore, the RCM is more general than the classical Fama


and MacBeth approach to aggregating a time series of cross-sectional estimates.

RCM may be stated as:


LPREM,


= at.


+ / (liquidity proxy),.l


+ error,


+ error,


where i is an index for funds, t is an index for time.


Variation in the coefficients


reflect industry wide random change in the level of premia as well as the changes in

the impact of liquidity on premia.

Table XV reports the results of the RCM analysis. All the coefficients on liquidity


proxies continue to be significant at


level.


4) Seemingly Unrelated Regression Models (SURI and SUR2)


The SUR models allow cross-equation correlation,


SUR models.


we examine two separate


The basic model is specified as a following system of equations:


LPREMi,


+ 3 (liquidity proxy);i,,


+ E.i








65

SURI accounts for the effects of omitted variables that may be common across funds.

SUR2 admits the notion that repeated observations on the same entity may produce


errors that are correlated across time.


The slopes and associated significance are


reported in


Table XV


The results on SUR2 may not be reliable because we were


constrained to using no more than ten weeks of data in SUR2 analysis (this limitation


is due to identification).


The results of SURI


(and most of SUR2) are substantively


similar to the results seen earlier.


In summary,


Table XV provides overwhelming evidence that the observed


relationship between liquidity and premia is unlikely to be an artifact of assumptions

in our model specification.


Some informal evidence


Are the Returns on Closed-end Funds Related to Low-Volume Stocks?


Lee et al. (1991) examine the relationship between premia and the return


behavior of the smallest decile of stocks on the NYSE.


Chen, Kan and Miller (1993)


take a second look at such a relationship and conclude that there is no special


relationship between premia and the small stocks. In similar spirit,


we explore the


whether the returns realized by closed-end funds are related to thinly traded (lowest

trading volume decile) securities in any special way. An existence of such a

rel2tinnvhin xvnnld cinooa ct th'sn tha nlvr'i'l-:,nir f'nnrte ih,-n o h~uoa Ii1- illirhi1irl








66

related to highly-traded securities.is Such evidence would be supportive of the

conjecture that liquidity of a fund may be a major determinant of the market value

and, therefore, may be related to premia in closed-end funds.

We tested the following time-series specification:


+ i Xi,


where t is an index for months spanning January


1963 to December


1991 and


is an


index for volume-decile portfolios of NYSE listed stocks.


Y, is the excess return realized by the closed-end fund industry in month t.


The excess


return was defined as the value-weighted return realized by the closed-end industry


minus the return on the value weighted market.


The stocks listed on NYSE (excluding


closed-end funds) were divided into ten portfolios based on their volume of trade in


the prior month. X, denotes value-weighted returns realized by the i1' decile in


excess


of the return realized on the value-weighted market, in month t.

Table XVI shows that the lowest volume decile has the strongest relationship


with the excess returns realized by the closed-end industry.


The R2 for the lowest


decile is about 34% compared to


for the smallest size decile and 0.33


for the


largest volume decile. Further, the R2 declines almost monotonically as we look at

higher deciles. Inclusion of dummies for January did not substantively affect the


results reported in


Table XVI.


Overall, the results in


Table XVI strongly suggest that


returns on closed-end funds may co-move with returns on low-volume securities.


This


= a;








67

result is surprising because, as mentioned above, the fundamentals of closed-end

funds are related to high-volume securities and not as much to low-volume securities.

Thinness of trading, despite the fundamentals, seems to affect the return behavior of

closed-end funds; and to this extent the results are consistent with the notion that

market values of closed-end funds, and therefore premia, may be related to liquidity.


Summary


We suggest that illiquidity


or expected frictions in the trading process, may


influence premia/discounts observed in closed-end funds. Specifically,


we conjecture


that premia (discounts) are observed when claims issued by the fund are more (less)

liquid than the underlying assets.

The liquidity hypothesis is appealing, because it extends extant literature in


several important ways.


In particular, it is consistent with premia as well as discounts,


it predicts that average premia/discounts may be different for different types of funds

(equity versus bond) and it explicitly recognizes the trading frictions that may restrict

apparent arbitrage opportunities in presence of premia/discounts.

Our evidence strongly supports the liquidity conjecture. For example, funds

with higher liquidity, as measured by proxies for trading activity and volatility, have


higher premia (or lower discounts) than funds with lower liquidity.


is observed in equity funds as well as in bond funds.


This relationship


Consistent with the liquidity








68
than in bond funds. Further, the observed relation between liquidity and premia is


robust to various assumptions about parameters and error structures.


Examination of


excess returns suggests that closed-end funds behave like illiquid securities, although

the funds hold highly liquid securities in their portfolios.

Overall, the liquidity hypothesis is supported in our data and may offer a

richer description of the premia/discount phenomenon in conjunction with existing

hypotheses.









69

TABLE [X
DESCRIPTIVE STATISTICS OF THE SAMPLE.


CLOSED-END EQUITY


JANUARY


AND BOND FUNDS LISTED ON THE NYSE


TO DECEMBER


Fnnn January 4, 1988, log of


(LDOL1,),


size (LSZ,,), log of volume (LVOL,,), log of dollar volume of trade


log of shares outstanding (LSH,,), loh of turnover rate (LTURN,,), lo of volatility (LSIG,,) and log of


premium


(LPREM,,) are calculated or each fund mon


a weekly


basis for about


208 weeks.


Means and standard


deviations, in parentheses, of variables are r:cpoted above.


Equity Funds Bond Funds


LPREMI -0.12 (0.10) -0.01 (0.07)


LSZ, 19.07 (1.25) 18.77 (0.95)


LVOL, 11.00 (1.58) 11.31 (1.56)


LDOLi, 13.64 (1.46) 13.65 (1.37)


LSHI 16.44 (1.25) 16.44 (1.07)


LSIG, -3.85 (0.65) -3.86 (0.56)


LTURN,, -0.83 (0.79) -0.52 (0.74)


NUMBER OF FUNDS 18 90













TABLE


AVERAGE SLOPES OF CROSS-SECTIONAL REGRESSIONS.
CLOSED-END EQUITY FUNDS LISTED ON THE NYSE:


JANUARY


TO DECEMBER 1991


From January 4


1988, log of


size (LSZ,,),


(LVOL


of dollar volume


trade (LDOL,,),


log of shares outlsiand ing


(LSH,,), log of turnover rate


(LTURN,), log


volatility (LSIG,,) and log


of premium


(LPREM,)


are calculated for"


each fund on a weekly basis.


The sample


is divided into twenty periods of about 10


weeks each


(partial pooling). Premia/di


SCmUntl.S,


expressed


as LPREM,,,


are regressed


on the explanatory


variables in


each of the twenty periods. This provides


20 crcss-sectiConal


regressions


on the semi-poo)led sample, or


one for each


period. Using the classical Fama-MacBeth appr


)ach, the


average


sh lpsc


are the time-se


ines averages


o, the slopes


obtained from the


20 regressions. The L-statistic


is simply


slope divided by its


time-series


standard


across


the 20 regressions.


basic cross-sectional


LPREM,,


irssion


+ h, X,,


LVOL,.,1 LDOL., 1 LSZ,., LSH., LTURN,., I LSIG,.,


0.008 (7.18)



0.02 (15.77)



0.022 (10.73)



0.006 (4.02)



0.019 (4.03)



-0.028(-7.48)


error


loh, of volume










71

TABLE XI
AVERAGE SLOPES OF CROSS-SECTIONAL REGRESSIONS.


CLOSED-END BOND FUND


JANUARY


LISTED ON THE NYSE


1988 TO DECEMBER


From January 4, 1988, log


size (LSZ,,), lor of volume (LVOL


,), log


of dollar


volume


of trade


(LDOL,),


log of shares outstanding (LSH,1), log


of turnover rate


(LTURN,,), log


of volatility


(LSIG,,) and log


premium (LPREM,,)


are calculated for each fund on


a weekly basis.


The sample


is divided into twenty periods of


about 10 weeks each (partial pooling). Premia/discounts,


expressedC


as LPREM,,,


are regressed


on the explanatory


variables each period. This provides


20 estimates from the semi-pooled sample, one for each period. Using the


classical Fama-Macbeth approach, the


average


are the time


series averages of the slopes obtained from the 20


regressions.


The (-statistic


is simply the


average


slope divided by


its time


series


standard


error


ac ross


the 20


regressions.


The basic cross-sectional


LPREM,,


regression is:


+ h, X,


LVOL,.,, LDOL,1 LSZ., LSH,,, LTURN..II LSIG,.1


0.007 (3.01)



0.011 (5.05)



0.022 (1 1.31)



0.014 (5.21)



-0.000 (-0.040)



-0.015(-3.98)











TABLE XII


LATENT VARIABLE MODELS


AVERAGE SLOPES OF CROSS-SECTIONAL


ANALYSES.


CLOSED-END EQUITY FUND
JANUARY 1988 TO DECENT


LISTED ON NYSE:


4BER 1991


From January 4,


1988.


log of


size (L


SZi),


og of volume (LVOL1i),


og of


dollar volume


of trade


(LDOLI,),


log iof shares outstanding


LSHI,),


log of


turnover


rate (LTURN;,),


of volatilit


(LSIG,) and lo


of premium


LPREM1,) are


calculated for each fund on a weekly basis. The sam


of about 10 weeks each. A


ple is divided into twenty periods


system of equations is estimated every period


provides


20 estimates from the


semi-pooled samp


one for each period.


Using the


classical Fama-Macbeth approach,


the sl


the average slopes


opes obtained from the 20 regressions.


are the


t-statistic


time-series averages o


is simply the average slope


divided by its time-series standard error across the 20 regressions.


The basic


stem


of equations is:


LPREM,


= P,


41) + 0,1,


4i) + ,,,


(4.) + 0 j,


P= 34l (


where, i=.


to 18


equity funds


and t=


to 20 periods.


2. Y3 and Y4 are


some proxies that measure unobservah


uidity (4) with


error (6).


These


equations


are estimated simultaneously


as a sy


stem with


) and 0 as parameters


estimates are consistent [See Bollen


1989)


for details


liquidity proxies are











TABLE XII


(CONTINUED)


LATENT VARIABLE MODELS


AVERAGE SLOPES OF CROSS


-SECTIONAL ANALYSES.


CLOSED-END EQUITY FUNDS LISTED ON NYSE:


JANUARY


1988 TO DECEMBER


Each row


in the


above shows estimates


karate mod


of simultaneous


equations, and each column (within a given row


cell entries represent estimates of coeffi


is a separate equation within the model.


clients (or estimates of /3) and t-values of estimates are


shown in parentheses


LPREM LVOL LDOL LSZ LSH LTURN LSIG
0.023 (9.77) 0.99 (12.30) 1.52 (12. I1)
0.014 (6.83) 1.89 (9.37) 1.08 (8.10)
0.023 (7.76) 1.17 (7.86) 0.58 (12.12)
0.031 (12.42) 1.50 (19.08) 1.05 (26.57)
0.022 (9.21) 2.40 (11.51) 0.68 (13.23)
0.040 (11.93) 1.36 (7.00) 0.35 (10.18)
0.0.090 0.72 (4.14) -0.15 (-6.23)
0.083 (5.98) 0.57 (3.83) 0.13 (1.77)
0.062 (5.45) 0.75 (5.48) -0.23 (-7.13)
0.026 (4.64) 0.51 (3.90) 0.26 (0.87)
0.022(6.49) 1.14 (4.88) 1.54 (9.26) -0.052(-2.88)
0.010 (2.91) 1.92 (8.84) 1.03 (7.31) 0.05 (2.96)
0.011 (2.67) 0.83 (5.42) 0.93 (9.80) 0.14 (7.01)
0.029 (7.84) 1.55 (10.93) 1.06 (12.57) -0.041(-1.98)
0.022 (9.60) 2.14 (16.98) 0.73 (18.56) 0.001(0.12)
0.055 (4.48) 0.77 (5.27) 0.48 (2.87) -0.044(-1.09)
0.079 (3.78) 0.50 (5.22) -0.07 (-1.02) -0.27(-4.72)
0.006 (0.53) 0.20 (2.91) 0.70 (2.95) 0.05 (0.49)








74

TABLE XIII
LATENT VARIABLE MODELS. AVERAGE SLOPES OF


CROSS-SECTIONAL ANALY


SES.


CLOSED-END BOND FUNDS LISTED ON NYSE:


JANUARY 1988 TO DECEMBER 199


From January 4, 1988, log of


volume of trade (LDOL;J,


size (LSZ;,), lo


log of shares outstanding (LSH;,),


'olume (LVOLt), log of dollar

log of turnover rate (LTURN1),


log of volatility (LSIG,,) and log of premium (LPREM,,) are calculated for each fund on a


weekly basis.


The sample is divided into twenty periods of about


0 weeks each. A system of


equations is estimated every period.


This provides 20 estimates from the semi-pooled sample,


one for each period. Using the classical Fama-MacBeth approach, the average slopes are the


time-series averages of the slopes obtained from the 20 regressions.


The t-statistic is simply


the average slope divided by its time-series standard error across the 20 regressions.

The basic system of equations is:


LPREM,,
Y2-,

Y31,

Y4 =


= (40) + 0i


= (40 + 02i,


= f41


<4,) + 0.6,


where, i=


to 90 bond funds and t=


to 20 periods.


Y3 and Y4 are some


proxies that measure unobservable liquidity (4,) with error (0).


These equations are estimated


simultaneously as a system with P, 4P and 0 as parameters and the estimates are consistent


[See, Bollen (1989) for details.


The liquidity proxies are lagged by one week in the


estimation process.











TABLE XIII


LATENT VARIABLE MOD


(CONTINUED)
'ELS. AVERAGE SLOPES OF


CROSS-SECTIONAL ANALYSES.
CLOSED-END BOND FUNDS LISTED ON NYSE:


JANUARY


1988 TO DECEMBER 199


Each row


n the table ahov


shows


estimates


of a separate model


of simultaneous


equations, and each column


within a given row


is a separate equation within the model.


cell entries represent estimates of coeffi


clients (or estimates ot' p) and t-values of estimates are


shown in parentheses.


LPREM LVOL LDOL LSZ LSH LTURN LSIG
0.024 (7.15) 1.12 (9.55) 1.26 (7.80)
0.022 (6.39) 1.23 (13.16) 1.33 (11.61)
0.009 (1.81) 1.53 (5.59) 0.69 (7.53)
0.012 (2.54) 1.31 (5.71) -0.29 (-6.74)
0.020 (6.09) 1.30 (13.16) 1.10 (13.84)
0.026 (10.00) 1.27 (5.25) 0.24 (1.35)
0.025 (7.56) 1.39 (6.05) -0.19 (-8.21)
0.037 (9.00) 0.90 (5.20) 0.27 (8.07)
0.029 (9.37) 0.84 (6.81) -0.15(-12. 15)
0.025 (4.61) 1.04 (4.32) 0.64 (2.89)
0.020 (4.95) 0.96 (5.12) -0.31 (-3.85)
0.019 (3.98) 1.51 (18.15) 0.82 (13.92) -0.17(-10.71)
0.013 (3.14) 1.55 (22.58) 1.00 (24.90) -0.16(-10.16)
0.012 (3.01) 1.73 (15.07) 0.48 (9.73) -0.14(-7.27)
0.022 (6.92) 1.17 (15.73) 1.01 (13.93) -0.12(-5.72)
0.017 (5.37) 1.24 (42.83) 1.07 (63.83) -0.16(13.76)
0.019 (5.27) 0.86 (2.35) 0.21 (1.73) -0.11(-4.03)
0.035 (9.83) 0.86 (8.85) 0.27 (7.96) -0.15(-9.44)
0.020 (4.62) 1.14 (10.55) 0.40 (4.49) -0.15 (-8.57)










76

TABLE XIV


BINOMIAL LOGIT


ANALYSIS OF THE PROBABILITY OF OBSERVING


PREMIA/DISCOUNTS IN
CLOSED-END FUNDS LISTED ON THE NYSE:


JANUARY 1988 TO DECEMBER


Premia Discount Difference


Equity Fund 0.18' 0.82* -0.64'



Bond Fund 0.53' 0.47* 0.06


Difference -0.35* 0.35*


* denotes significance at


Probabilities of premia/discount, given fund


Binomial logit specification


level.


ype, are shown above


is used to estimate the probability of observing prem ia or discounts given the


type of fund (equity


or bond). The


salmp


Il consists


of weekly observations, 'f1 1/4/88


to 12/31/91, on


18 equity


lifunds and 90 bond funds


are listed on the NYSE.


Ilnie-series


and cress-scction, data


is used in this


analysis


The basic


log odds nKmodel


Log (Y,/ -Y,)


Where, i


is an index


for observe


tionls.


= 1 for premil f 0 lor discounts.


X, denotes the type


of fund,


I for equity


fund or


=2 for bond fund.


= 0- +0













TABLE


SENSITIVITY


CLOSED-END EQUITY FUND


JANUARY


ANALYSIS.


LISTED ON NYSE:


TO DECEMBER


* SUR2 analysis


not reliable


hc aluse


it had


to bc limited it teLn


weeks of dtta for the purpose


identification. Small


sample properties


of SUR


estimates


are not appealing.


From January 4,


1988. log of size (LSZ,,),


IOg of


volume


(LVOL


). log


of dollar


volume


of trade


(LDOL,),


shares


outstanding (LSH,,), log


turnover


rate (LTURN,,).


hlI of


volatilit


y (LSIG,,) and log


premium (LPREM,,)


are calculated for


each fund on


a weekly


basis for 208 weeks.


Five different


economy it etric


techniques


are used


to study the


rohustlness of the relItionship between


liquidity


and premia.


Fixed Effects Model (FIM), Random Effects Model


(REIM),


Coefficients


Model (RCM)


seemingly unrelated


regresstions


models (SURI


,SUR2)


were used.


dependent


variable


is premium (LPREM).


The general


llmodel i


LPREM,


= t,, + Y, (liquidity


proxy),.


where i is an index lor rund


and I is


an index for"


Different


restrictions


ion paIfIrameters


errors


are considered by


the five


Models.


Sec section


IV for details IV.


Cocfficients


(B) and


abIove is a stand


iated t-values


Iare shown


Each cell


alone


result. Each


LVOL LDOL LSZ LSH LTURN LSIG


FEM 0.010(8.33) 0.023(19.18) 0.023(16.05) 0.007(4.77) 0.023(9.30) -0.036(-11.56)


REM 0.010(8.39) 0.023(19.35) 0.023(16.31) 0.007(4.70) 0.022(9.44) -0.027(-9.64)


RCM 0.010(7.14) 0.022(16.58) 0.021(14.98) 0.006(3.85) 0.028(8.58) -0.031(-6.79)


SUR1 0.009(21.53) 0.022(43.98) 0.024(37.06) 0.006(9.18) 0.022(23.37) -0.004(-2.81)


SUR2' 0.003(2.24) 0.014(10.72) 0.008(2.68) -0.002(-1.14) 0.011(9.92) -0.010(-7.70)


row


tassoCe












TABLE


TIME-SERIES ANALYSIS OF RELATION BETWEEN MONTHLY EXCESS RETURNS
ON VOLUME DECILE PORTFOLIOS AND CLOSED-END FUNDS LISTED ON NYSE.


JANUARY 1963 TO DECEMBER


Results of twelve sepanirte


excess


line-series


return on all closed-end funds in month t


ressionls


is value-weighted


, are shownll helow. Y,


excess returns,


in month I,


: value-weighted
on the i volume


decile portfolio of the NYSE listed stocks (excluding closed-end


funds). The


excess


returns


arc calculated


as value-


weighted returns on a portfolio minus the returns on value-weighted market.


The basic model


= a, + 0, X,,


This equation


is estimated separately for


each docile, i,


over Ui


tlime-series


of 348 months (t).


Coefficients,t,, and associated t-values (in parentheses) are shown above.


Volume Decile, i /3, R %

1 (smallest volume) 0.52 (13.34) 34%

2 0.38 (9.20) 20%

3 0.34 (7.48) 14%

4 0.31 (6.37) 11%

5 0.30 (5.55) 8%

6 0.22 (3.48) 3%

7 0.31 (4.12) 5%

8 0.24 (3. 18) 3%

9 0. 11 (1.33) 0.5%

10 (largest volume) 0.08 (1.07) 0.33%

Size Decile 1 0.16 (6.86) 12%

Size Decile 10 -0.87 (-4.85) 6.4%



















APPENDIX A: PROPERTIES OF PORTFOLIOS


PROPERTIES OF TEST PORTFOLIOS IN TERMS OF


AVERAGE YEARLY RETURNS,


YEARLY-AVERAGE TRADING VOLUME, AVERAGE MARKET CAPITALIZATION


AND THE AVERAGE NUMBER OF FIRMS


JULY


IN A GIVEN PORTFOLIO:


1963 TO JULY


Four types


of portfolios of NYSE listed firms


are flnned and examined, the rules for construction


of the


portfolios are described below:


1) Twenty


size based portfolios


are


ionned


on July


every


car, chased on


the market capitalization


of June'


30 of the same year. Portfolio


an equally


weighted portfolio


of the smallest


of the NYSE listed finns


a given year.


Portfolio 20 represents an equally weighted pofolflio


of the lir"est


of the


NYSE listed finns in


vel year.


The properties of these portfolios


are prese


ntccd in panel


A. Average retlums are the time


series


average


yearly


returns (percentage change in the value of one dollar


of investment) on these portf li)os


over the 28


period.


Average


Size is the time series


s average


of size


a portlfilio;


where the


a portfolio is an equally


weighted


average


of the natural logarithm of market capitalizations


of finns included in


a portfolio.


Average


trading


volume is a tlie series average


of the trading


volume for a portfolio;


where


the trading


volume for


weighted


average


of the natu nral logarithm of movingv


average


of past


year s


volume for finns included in


portfolio.


wenty volume based portfolios are ftormd and


examined


using


a protcedurc similar to the


one above


except


the basis foir finni


ng portfolios


volume instead


i)1 S1"


Volume


is calculated


using


the past


year's


average


trading


volume


a linl and this number


is used


to divide


NYSE


listed linns


intt tvweinty groups.


3) Thirty size-volume based portfolios


are li 11ned


every


,as follows: On July'


1 of e


very year ten


based portfolios


ar lionned as described in I) above.


Each of these


ten portfolios


is further


subdivided into


portfolios based on past year's


average trading


volume flor each


of the member finns in a gi


ven size gr


iup. The


breakpoints for volume are calculated by dividing all the NYSE listed finlls into three groups on the basis (oi trading


a portfolio











4) Thirty volume-size based portfolios


are lnired similar to the size-volume portfolios except the firmnns


first ranked on


volume and


then on


Types of Portfolios used in Pane


Panel A:


Panel B:


Panel C:


Panel [

Panels


A throu


size based portfolios


20 volume based portfolios


size-volume based portft


):


volume-size based portfolios


A through D are on the following four separate pages.


size.












APPENDIX A: PANEL A
SIZE BASED PORTFOLIOS.


TWENTY PORTFOLIOS ARE FORMED EVERY YEAR OVER A


28 YEAR PERIOD.


THE TIME SERIES AVERAGES OF PROPERTIES OF THESE PORTFOLIOS ARE


SHOWN BELOW


: TIME PERIOD JULY-1963 TO JULY-1990.


Portfolio Average Returns % Avg.Ln(Size) Avg.Ln(volume) Finns

I (small) 17.84 16.25 10.85 54.64

2 18.26 17.03 11.04 55.07

3 18.79 17.42 11.18 55.21

4 15.46 17.70 11.38 55.07

5 18.92 17.96 11.47 54.93

6 16.39 18.20 11.61 55.29

7 17.96 18.42 11.78 55.14

8 18.29 18.64 11.87 55.07

9 16.24 18.86 12.04 55.25

10 15.69 19.07 12.14 54.86

11 14.17 19.27 12.30 55.29

12 13.65 1949 12.49 55.21

13 14.68 19.71 12.61 55.11

14 14.75 19.95 12.77 55.14

15 14.98 20.20 12.96 55.04

16 12.36 20.43 13.12 55.14

17 13.20 20.70 13.27 55.11

18 11.81 21.00 13.43 55.21

19 10.92 21.40 13.69 55.07









82

APPENDIX A: PANEL B
VOLUME BASED PORTFOLIOS.
TWENTY PORTFOLIOS ARE FORMED EVERY YEAR OVER A 28 YEAR PERIOD.
THE TIME SERIES AVERAGES OF PROPERTIES OF THESE PORTFOLIOS ARE


SHOWN BELOW


: TIME PERIOD JULY


-1963 TO JULY


-1990.


Portfolio Average Returns Avg.Ln(Size) Av,.Ln(volume) Finns

1 (low volume) 17.63 17.47 9.71 54.64

2 17.37 17.68 10.41 55.07

3 17.72 17.87 10.77 55.25

4 18.22 8.06 11.05 55.04

5 17.32 18.26 11.28 55.04

6 16.22 18.36 11.49 55.18

7 17.14 18.53 11.69 55.18

8 15.42 18.68 11.88 55.04

9 14.79 18.86 12.07 55.25

10 15.65 19.05 12.25 54.89

11 15.21 19.24 12.43 55.25

12 16.05 19.41 12.61 55.21

13 16.29 19.67 12.79 55.14

14 13.78 19.80 12.96 55.10

15 14.09 19,93 13.14 55.04

16 14.13 20.15 13.34 55.14

17 13.55 20.34 13.56 55.21

18 12.69 20.54 13.81 55.21

19 12.49 20.82 14.12 55.07

20 (High Volume) 9.63 21.46 14.72 54.75









83


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APPENDIX B
TESTABLE HYPOTHESES: AN ALGEBRAIC ILLUSTRATION


Hypothesis #1


Within a group of funds that hold similar assets (equity or


bond), premia increase (discounts decrease) as the liquidity ffimunds'


increases.


Consider traded securities indexed on


with market value per share defined as


The perfect market value per share (value of cash flows promised by the security


in a perfect market) is denoted by Pi. Let us write the relationship between S, and P

as follows:


= P Fi*P =PI*(1-F)=P*L


where F, is a fraction of P, that is lost due to trading friction, and L, is a measure of


liquidity of security i. Equation (1) says that the market value (S,) of a security


equal to the perfect market value (P,) adjusted by some liquidity measure (L,); this a

statement that incorporates the value of liquidity.


Next, consider a closed-end fund that holds N; shares of security


in its portfolio.


can express the market value (MV,,,) and net asset value of the fund (NAV,,, ) in

terms of the fund's holdings as follows:


shares











NAVV, ,C


= -N


SS=I


ZN,


MV ud


= Lu


where equation (2) is an accounting identity and equation (3) is a restatement of


equation (1) in the context of a fund.


The liquidity of fund'


shares is denoted by


LI lund

From equations (2) and (3) we can write:


+premium


MVf.
NAV n,,


Ll=


where, ., is the weight of security i in the perfect market value of the fund's

portfolio. The law of large numbers says that as the number of securities in the fund


increases the weighted sum of liquidity


the population (u). Let


* L ,


will approach the mean liquidity of


be the mean liquidity of assets of type j (equity or bond).


Equation (4) suggests that within a given group of funds (equity or bond) the


denominator,


will be constant and premia increase as the numerator, Lrua,


increases; this is a statement of hypothesis #1.


Hypothesis #2:


A average premia/discounts may be different across different types


of funds (equity versus bond).


~ .. .. ...-- k -- ~ f ^ \ kL_ **/ *& -- ... .j r j .* ^ &L -- J- -\,k J^-n* .hk ... .' J-*.l Il, .-k, A .',J^. .. ,- .. .. ..^ f


y N. P.
-1 I











bars denote averages.


likely to be the same or L equity
"fund


traded in similar market structures.


In this expression the average liquidity of the fund shares is


= L hon, because they both are equity securities and are
liund


However, the mean liquidity of the bond assets is


likely to different from the average liquidity of the equity assets (or


I'equity assets


$bond :ssets) because they are traded in different market structures and to


this extent hypothesis #2 seems to be reasonable.














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BIOGRAPHICAL SKETCH


Vinay Datar was born and brought up in Bombay, India. He received the


National Merit scholarship, and graduated from high-school with


101' rank at the state


level. He received undergraduate degree in Mechanical Engineering from the Indian

Institute of Technology (I.I.T.). He had the first rank in the entrance examinations

(among over a million applicants at national level) for graduate level studies at I.I.T.


At the State University of New


York at Buffalo, he studied Mechanical Engineering


and Systems Analysis and left the graduate program before submitting the


dissertation.


He worked with Nashua Corporation on the Quality Program directed by


Edward Demming. As a co-founder of a start-up company, he raised about two


million dollars in venture capital, and later sold the company to the public. At Xerox

Corporation in Rochester, he was involved in Research and Development as a senior

Engineer, and later he joined the Ph.D. program in Finance at the University of

Florida.









certify tha1


have real this study and lima


acceptable standards of scholarly presentation


II lily op)lnlon


and is lu


y adequate,


conl form


in scope and


quality


as a dissertation for


degree of Doctor of P'1


losophy.


1<


Robert C.


~adcli lfe


Chaifhian


Associate Professor of Finance


certify tha


have


acceptable standards of scho
quality, as a dissertation for


ead this study and
early presentation


degree


in mily opinion it conlforms


and is fully adequate,


of Doctor of


in scope and


iloso


M iwL J. llannery
Prolfessor of financee


certify


accepltab


that I


standards


ave read


study


of scholarly presentation


I thai in imy opinion it cooforms to
and is fully adequate, in scope aUI


quality


as :I disserta


on for the


ree of


f Doctor of


'hilosoplhy


Aoilisoan H. reo [illon
Associate Professor of Economnics


certify


that I have read


this stu


dy and


acceptable standards of scholarly presentation


that in my opinion it conforms to
mnd is fully adequate, in scope and


Iiality


as a dissernl


ion for


ree of Doctor of Philosophy.


c\)2 tNJr^^rvvyyv


Nimalelndran


island Protfessor of Finance


disserta


oni was


he Graduate Faculty of the )Department of


Finance, Insurance and Real Estate in


College ol


Ihiusiness Admnii


the Graduate Schoo


and was accepted as partial fu! C


ment of


he requirements for


the degree of Doctor of Philosophy.


sIralion and to